,0 


ANALYTIC    GEOMETRY 


•V><y^° 


ANALYTIC    GEOMETRY 


FOR 

TECHNICAL  SCHOOLS  ANI)  COLLEGES 


P.    A.    LAMHERT,   M.A. 

INSTKUCTOli   IX    MATHEMATICS,    LEHIGH   UNIVERSITY 


THE    MACMILLAN    COMPANY 

LONDON:  MACMILLAN  &  CO.,  Ltd. 

1897 

All  rights  reserved 


,5 


QK^ 


Copyright,  1897, 
By   the   MACMILLAN   COMPANY. 


Norfaaati  19rfS3 

.1.  S.  Cushins  S;  Co.      Berwick  &  Smith 
Norwood  Mass.  U..S.A. 


PREFACE 

The  object  of  this  text-book  is  to  furnish  a  natural 
but  thorough  introduction  to  the  principles  and  applica- 
tions of  Analytic  Geometry  for  students  who  have  a 
fair  knowledge  of  Elementary  Geometry,  Algebra,  and 
Trigonometry. 

The  presentation  is  descriptive  rather  than  formal. 
The  numerous  problems  are  mainly  numerical,  and  are 
intended  to  give  familiarity  with  the  method  of  Analytic 
(ieometry,  rather  than  to  test  the  student's  ingenuity  in 
guessing  riddles.  Answers  are  not  given,  as  it  is  thought 
better  that  the  numerical  results  should  be  verified  by 
actual  measurement  of  figures  carefully  drawn  on  cross- 
section  paper. 

Attention  is  called  to  the  applications  of  Analytic 
Geometry  in  other  branches  of  Mathematics  and  Physics. 
The  important  engineering  curves  are  thoroughly  dis- 
cussed. This  is  calculated  to  increase  the  interest  of  the 
student,  aroused  by  the  beautiful  application  the  Analytic 
Geometry  makes  of  his  knowledge  of  Algebra.  The 
historical  notes  are  intended  to  combat  tlie  notion  that 
a  mathematical  system  in  all  its  completeness  issues 
Minerva-like  from  the  brain  of  an  individual. 

P.   A.    LAMBERT. 

80()5,';4 


TABLE   OF   CONTENTS 

ANALYTIC  GEOMETRY  OF   TWO  DIMENSIONS 

CHAPTER    I 

Rectangi'lar   Coordinates 
uniriE  i-AOK 

1.  Introduction 1 

2.  Coordinates 1 

3.  The  Point  in  a  Straight  Line 2 

4.  The  Point  in  a  Plane 3 

5.  Distance  between  Two  I'oints        .......  7 

6.  Systems  of  Points  in  the  Phme 8 

CHAPTER    II 
Equations  of  Geometric  Figures 


7.     Tlie  Straight  Line 

.     13 

8.    The  Circle 

.     15 

0.    Tlie  Conic  Sections 

.     15 

10.    The  Ellipse 

.     18 

11.    The  Hyperbola 

.     21 

12.    The  Parabola 

.     24 

CHAPTER    HI 
Plottixc;  ou  Ai.gki'.uaic  Equations 


13.  General  Theory 

14.  Locus  of  First  Degree  I'^piation 

15.  Straight  Line  through  a  Point 
IG.  Tangents         .... 


CONTENTS 


17.  Points  of  Discontinuity  . 

18.  Asymptotes 

19.  Miixiinum  and  Minimum  Ordinates 

20.  Points  of  Inflection 

21.  Diametric  Method  of  Plotting  Equations 

22.  Summary  of  Properties  of  Loci      . 


PAGE 

.  33 

.  34 

.  30 

.  37 

.  39 

.  39 


CHAPTER   IV 
PLOTTiNr,  OF  Transcendental  Equations 

Elementary  Transcendental  Functions 45 

Exponential  and  Logaritlimic  Functions 45 

Circular  and  Inverse  Circular  Functions 47 

Cycloids 54 

Prolate  and  Curtate  Cycloids 57 

Epicycloids  and  Hypocycloids 58 

Involute  of  Circle 59 


CHAPTER   V 


Tkansfokmation  of  Coordinates 

30.  Transformation  to  Parallel  Axes    . 

31.  From  Uectangular  Axes  to  Rectangular 

32.  <)bli(iue  Axes 

33.  From  Rectangular  Axes  to  Oblique 

34.  General  Transformation 

35.  The  Problem  of  Transformation    . 


CHAPTER   VI 


Polar  Coordinates 

30.  Polar  Coordinates  of  a  Point . 

37.  Polar  Equations  of  Geometric  Figures 

38.  Polar  Equation  of  Straiglit  Line     . 

39.  Polar  Equation  of  Circle 

40.  Polar  Equations  of  the  Conic  Sections 

41.  Plotting  of  Polar  Eciuations    . 

42.  Transformation  from  Rectangular  to  Polar  Coordinates 


CONTENTS 


CH AFTER   VII 
Propertiks  of  the  Straight  Line 

AKTICI.E 

43.  Equations  of  the  Straight  Line     . 

41.  Angle  between  Two  Lines   . 

4;").  Distance  from  a  Point  to  a  Line  . 

4<;.  E(iuations  of  Bisectors  of  Angles 

47.  Lines  through  Intersection  of  Given  Lines 

48.  Three  Points  in  a  Straight  Line    . 
40.  Three  Lines  through  a  Point 
50.  Tangent  to  Curve  of  Second  ( )rder 


PAGE 

81 
84 
85 
86 
87 
88 
89 
91 


CHAPTER    VIII 
Properties  of  the  Circle 

51.  Equation  of  the  Circle 93 

52.  Connnon  Chord  of  Two  Circles 94 

53.  Power  of  a  Point 95 

54.  Coaxal  Systems 97 

55.  Orthogonal  Systems 98 

56.  Tangents  to  Circles 1*'^ 

57.  Poles  and  Polars •         .102 

58.  Reciprocal  Figures 194 

59.  Inversion 196 


60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 


CHAPTER   IX 

Properties  of  the  Conic  Sections 

General  Equation HI 

Tangents  and  Normals H"^ 

Conjugate  Diameters 119 

Supplementary  Chords 122 

Parameters 1-'* 

The  Elliptic  Compass 12<> 

Area  of  the  Ellipse 127 

Eccentric  Angle  of  Ellipse 128 

Eccentric  Angle  of  the  Hyperbola 130 


CONTENTS 


CHAPTER   X 


Second  Degree  Equation 

ARTICLE  PAGE 

69.  Locus  of  Second  Degree  Equation 133 

70.  Second  Degree  Equation  in  Oblique  Coordinates        .        .         .  138 

71.  Conic  Section  through  Five  Points 141 

72.  Conic  Sections  Tangent  to  Given  Lines 142 

73.  Similar  Conic  Sections 144 

74.  Coufocal  Conic  Sections 146 


CHAPTER   XI 
Line  Coordinates 

75.  Coordinates  of  a  Straight  Line 149 

76.  Line  Equations  of  the  Conic  Sections 151 

77.  Cross-ratio  of  Four  Points 151 

78.  Second  Degree  Line  Equations 152 

79.  Cross-ratio  of  a  Pencil  of  Four  Rays 153 

80.  Construction  of  Projective  Ranges  and  Pencils  ....  155 

81.  Conic  Section  through  Five  Points 157 

CHAPTER   XII 

Analytic  Geoaikti;y  of  thi;   Complkx  Yari 


82.  Graphic  Rcpioscntation  of  the  Cuniiilex  Variable 

83.  Arithmetic  Operations  applied  to  Vectors   . 

84.  Algebraic  Functions  of  the  Complex  Variable     . 

85.  Generalized  Transcendental  Functions 


160 
162 
165 
168 


ANALYTIC   GEOMETRY  OF   THREE   DIMENSIONS 

CH.VPTER    XIII 

Point,  Line,  and  Plane  in  Space 

86.  Rectilinear  Space  Coordinates 171 

87.  Polar  Space  Coordinates 173 

88.  Distance  between  Two  Points 174 


CONTENTS 


89.  Equations  of  Lines  in  Space 

90.  Equations  of  the  Straight  Line  . 
9L  Angle  between  Two  Straight  Lines 

92.  The  Plane 

03.  Distance  from  a  Point  to  a  Plane 

94.  Angle  between  Two  Planes 


CHAPTER   XIV 


Curved    Sukfaces 


95. 

90. 

97. 

98. 

99. 
100. 
101. 
102. 


103. 
104. 
105. 
100. 
107. 
108. 
100. 
110. 
111. 
112, 
113. 


Cylindrical  Surfaces    . 

Conical  Surfaces 

Surfaces  of  Revolution 

The  Ellipsoid 

The  Hyperboloids 

The  Paraboloids  . 

The  Conoid 

Equations  in  Three  Variables 

CHAPTl 


Second  Degree   Equation  in  Three  Variables 


Transformation  of  Coordinates 

Plane  Section  of  a  Quadric 

Center  of  Quadric 

Tangent  Plane  to  Quadric  . 

Reduction  of  General  Equation  of  Quadric 

Surfaces  of  the  First  Class 

Surfaces  of  the  Second  Class 

Surfaces  of  the  Third  Class 

Quadrics  as  Ruled  Surfaces 

Asymptotic  Surfaces   . 

Orthogonal  Systems  of  (Juadrics 


R   XV 


ANALYTIC   GEOMETRY 


CHAPTER   I 

EEOTANGULAK  COORDINATES 

Art.  1. — Introduction 

The  object  of  analytic*  geometry  is  the  study  of  geometric 
figures  by  tlie  processes  of  algebraic  analysis. 

The  three  fundamental  problems  of  analytic  geometry  are: 

To  find  the  equation  of  a  geometric  figure  or  the  e(;[uations 
of  its  several  parts  from  its  geometric  definition. 

To  construct  the  geometric  figure  represented  by  a  given 
equation. 

To  find  the  relations  existing  between  the  geometric  prop- 
erties of  figures  and  the  analytic  properties  of  equations. 

Art.  2.  —  Coordinates 

Any  scheme  by  means  of  which  a  geometric  figure  may  be 
represented  by  an  equation  is  called  a  system  of  coordinates. 

*  The  reasoning  of  pure  geometry,  the  geometry  of  Euclid,  is  mainly 
synthetic,  that  is,  starting  from  something  known  we  pass  from  conse- 
(luence  to  consequence  until  something  new  results.  The  reasoning  of 
algebra  is  analytic,  that  is,  assuming  what  is  to  be  demonstrated  we  pass 
from  consequence  to  consequence  until  the  relation  between  the  unknown 
and  the  known  is  found.  The  term  "analytic  geometry"  is  therefore 
equivalent  to  algebraic  geometry.  The  application  of  algebra  to  the  de- 
termination of  the  properties  of  geometric  figures  was  invented  by 
Descartes  (1596-1050),  a  French  philosopher,  and  published  in  Leyden 
in  1G37. 


2  ANALYTIC  GEOMETRY 

The  coordinates  of  a  point  are  the  quantities  which  deter- 
mine the  position  of  the  point. 

Along  the  line  of  a  railroad  the  position  of  a  station  is 
determined  by  its  distance  and  direction  from  a  fixed  station ; 
on  our  maps  the  position  of  a  town  is  determined  by  its  lati- 
tude and  longitude,  the  distances  and  directions  of  the  town 
from  two  fixed  lines  of  the  map ;  the  position  of  a  point  in  a 
survey  is  determined  by  its  distance  and  bearing  from  a  fixed 
station. 

On  these  different  methods  of  determining  the  position  of 
a  point  are  based  different  systems  of  coordinates. 

Akt.  3.  —  The  Point  in  a  Straight  Line 

On  a  straight  line  a  single  quantity  or  coordinate  is  sufficient 
to  determine  the  position  of  a  point.     Let  0  be  a  fixed  point 

-8    -7  .-6    -5  -4    -3-2-10       1        2       3       4       5      6       7       8 
Fio.  1. 

in  the  line;  adopt  some  length,  such  as  01,  as  the  linear  unit; 
call  distances  measured  from  0  towards  the  right  positive,  dis- 
tances measured  from  0  towards  the  left  negative.  Let  a 
point  of  the  line  be  represented  by  the  number  which  ex- 
presses its  distance  and  direction  from  the  fixed  point  0. 
Then  to  every  real  number,  positive  or  negative,  rational  or 
irrational,  there  corresponds  a  definite  point  in  the  straight 
line,  and  to  every  point  in  the  line  there  corresponds  a  definite 
real  number.  This  fact  is  expressed  by  saying  that  there  is 
a  "  one-to-one  correspondence  "  between  the  points  of  the  line 
and  real  numbers. 

The  algebra  of  a  single  real  variable  finds  a  geometric  inter- 
pretation in  the  straight  line.  Denoting  by  x  the  distance 
and  direction  of  a  point  in  the  straight  line  from  0,  that  is 
letting  X  denote  the  coordinate  of  the  point,  the  equation 
cc^  —  2a;  —  8  =  0  locates  the  two  points  (4),  (—  2),  in  the 
straight  line. 


UECT ANGULAR   COORDINATES  3 

Problems.  —  1.  Locate  in  the  straight  Hue  the  points  3;  —2;  1^  ; 
-2.5;   -5;  f. 

2.   Locate  VG  ;   -VS;   VlO ;   VT. 

Suggestion. — The  numerical  value  of  VS  can  be  found 
only  approximately.      The  hypotenuse  of   a  rit;ht  triangle        /^ 
whose  two  sides  about  the  right  angle  are  2  and  1,  repre-        '^/ 
sents  v'5  exactly.  I 

3.  rind  the  point  midway  between  xj  and  x^.  Fig.  2. 

4.  Find  the  point  dividing  the  line  from  Xi  to  X'2  internally  into  seg- 
ments whose  ratio  is  /•. 

5.  Find  the  puiiit  dividing  the  lino  from  Xi  to  X2  externally  into  seg- 
ments wiinsi'  ratio  is  r. 

6.  Locatt!  the  roots  of  a;2  +  2  a;  -  8  =  0. 

7.  Locate  the  roots  of  xr  —  i  x  —  i  =  0. 

8.  Locate  the  roots  of  x^  -  0  x-  +  11  x  -  0  =  0. 

9.  Find  the  points  dividing  into  three  equal  parts  the  line  from  2 
to  14. 

10.  Find  the  points  dividing  into  three  ecjual  parts  the  line  from  X] 
to  X.J. 

11.  Find  the  point  dividing  a  line  8  feet  long  internally  into  segments 
in  the  ratio  3  :  4. 

12.  A  uniform  bar  10  feet  long  has  a  weight  of  15  pounds  at  one  end, 
of  25  pounds  at  the  other  end.  Find  the  point  of  support  for  equilib- 
rium. 

Al^T.  4.  —  Thk  Poikt  in  a  Plane 

To  determine  the  position  of  a  point  in  a  plane,  assume  two 
straight  lines  at  right  angles  to  each  other  to  be  fixed  in  the 
plane.  These  lines  are  called  the  one  the  A'-axis,  the  other 
the  F-axis.  The  distance  from  a  point,  in  llic  plane  to  either 
axis  is  moasiired  on  a  line  ]):ira]l('l  to  tlie  other  axis;  the 
direction  of  the  point  I'roui  tlic  axis  is  indicated  by  the  alge- 
braic sii^n  prchxcd  to  the  nniiilicr  expressing,'  tlie  distance  from 
tlio  axis. 


4  ANALYTIC  GEOMETRY 

Distances  measured  parallel  to  the  X-axis  to  the  right  from 
the  l''-axis  are  called  positive  ;  those  measured  to  the  left  from 
the  F-axis  are  called  negative.  The  distance  and  direction  of 
a  point  from  the  T-axis  is  called  the  abscissa  of  the  point,  and 
is  denoted  by  x. 


+ 

Y 

( 

r3,2) 

G,2) 

-X 

A 

+  X 

■ 

( 

-3-2) 

(3-2) 

-Y 

Distances  measured  parallel  to  the  F-axis  upward  from  the 
X-axis  are  called  positive;  those  measured  downward  from 
the  X-axis  are  called  negative.  The  distance  and  direction  of 
a  point  from  the  X-axis  is  called  the  ordinate  of  the  point,  and 
is  denoted  by  y. 

The  axes  of  reference  cut  the  plane  into  four  parts.  Calling 
the  part  +X^+F  the  first  angle,  +F^4~X  the  second  angle, 
-XA~  Y  the  third  angle,  ~  F^l+X  the  fourth  angle,  it  is  seen 
that  in  the  first  angle  ordinate  and  abscissa  are  both  positive ; 
in  the  second  angle  the  ordinate  is  positive,  the  abscissa  nega- 
tive ;  in  the  third  angle  ordinate  and  abscissa  are  both  negative ; 
in  the  fourth  angle  the  ordinate  is  negative,  the  abscissa  posi- 
tive. 

The  a1)scissa  of  a  point  determines  a  straight  line  parallel  to 
the  l''-axis  in  which  the  point  must  lie.  For,  by  elementary 
geometry,  the  locus  of  all  points  on  one  side  of  a  straight  line 


RECTANG  ULA  li   COORD  IN  A  TES 


ami  equidistant  from  the  straight  line  is  a  straight  line  parallel 
to  the  given  line. 

The  ordinate  of  a  point  determines  a  straight  line  parallel  to 
the  X-axis  in  which  the  point  must  lie. 

If  both  ordinate  and  abscissa  of  a  point  are  known,  the  point 
must  lie  in  each  of  two  straight  lines  at  right  angles  to  each 
other,  and  must,  therefore,  be  the  intersection  of  these  lines. 
I  lence  ordinate  and  abscissa  together  determine  a  single  point 
in  the  plane. 

Conversely,  to  a  point  in  the  plane  there  correspomls  one 
ordinate  and  one  abscissa.  For  through  the  point  only  one 
straight  line  parallel  to  the  I'-axis  can  be  drawn.  This  fact 
determines  a  single  value  for  the  abscissa  of  the  point. 
Through  the  given  point  only  one  parallel  to  the  X-axis  can 
be  drawn.  This  determines  a  single  value  for  the  ordinate  of 
the  i)oint. 

The  abscissa  and  ordinate  of  a  point  as  defined  are  together 
the  rectangular*  co- 
ordinates of  the  point. 
The  point  whose  co- 
ordinates are  x  and  y 
is  spoken  of  as  the 
point  (.r,  ?/).  There  is 
a  "  one-to-one  corre- 
spondence "  between 
the  symbol  (x,  y)  and 
the  points  of  the 
Xl'-plane. 

Problems.  —  1.    Locate  the  point  (3,  —  4). 

Lay  off  ;]  linear  units  on  the  X-axis  to  the  right  from  the  origin,  and 
thrru  is  found  the  straight  line  parallel  to  the  3'-axis,  in  whicli  the  point 
must  lie.  On  this  line  lay  off  4  linear  units  downward  from  its  intersec- 
tion with  tlie  J-axis,  and  the  point  (3,  —  4)  is  located. 

*  This  method  of  representing  a  point  in  a  plane  was  invented  by  Des- 
cartes.    Hence  these  coordinates  are  also  called  Cartesian  coordinates. 


Y 

A 

'' 

( 

3,-4) 

1 

6 


ANA L  YTIC   GEOMETli  Y 


2.  Locate  (-3,0);  (0,4);  (1,  -1);  (-1,-1);  (-7,5);  (10,-7); 
(15,  20). 

3.  Locate  (2i^3);   (-  1,  SJ);   (%  -  51)^(7.8,  -  4.5). 
Locate  (V2,  \/5);  (-Va,  Vl7);   (V50,  V75). 
Construct  the  triangle  whose  vertices  are  (4,  5),  (  —  2,  7), 
Find  the  point  midway  between  (4,  7),  (0,  5). 
Find  the  point  midway  between  (x',  y'),  (a;",  y"). 
Find  the  area  of  tlie  triangle  whose  vertices  are  (0,  0),  (0,  8),  (0,0). 
Find  the  area  ol  the  triangle  whose  vertices  are  (2,  1),  (5, 4),  (9, 2). 


4. 


1. 


-3,  -6). 


Y 

(5 

4) 

/ 

\ 

/ 

^ 

::^ 

(-97-2 

1 

>  1  "l 

Z 

L- 

- 



■ 

a' 

" 

(2:o) 

(5 

0) 

(9i0) 

Fio.  5. 
Suggestion.  — The  area  of  the  triangle  is  the  area  of  the  trapezoid 
wliose  vertices  are  (2,  1),  (2,  0),  (5,  4),  (5,  0),  plus  the  area  of  tlie 
trapezoid  whose  vertices  are  (5,4),  (5,  0),  (9,  2),  (9,  0),  minus  the  area 
of  the  trapezoid  whose  vertices  are  (2,  1),  (2,  0),  (9,  2),  (9,  0). 

10.  Show  that  double  the  area  of  the  triangle  whose  vertices  are 
(xi,  2/i),  (X2, 2/2),  (X3,  2/3)  is  2/1  (a-3  -  .r2)+  2/2(a'i  -  a^3)+  2/3(^2  -  .^i). 

11.  Show  that  double  the  area  of  the  quadrilateral  whose  vertices  are 
(.1-1, 2/1),  (a^2, 2/2),  (scs,  2/3),  {Xi,  2/4)  is  yx{Xi  -  X2)  +  2/2(xi  -  0:3)  +  2/3(0:2  -  Xi) 
+  2/4(*'3  -  a^i). 

12.  Show  that  double  the  area  of  the  pentagon  whose  vertices  are 
(a;i,  2/1),  (3^2,  2/2),  ('^3,  2/3),  (X4,  2/4),   (A-5,  2/5)  is  2/1  (a-5  -  0:2)  +  2/2(^1  -  s's) 

+  2/3(^2  -  3:4)  +  2/4(«3  -  X5)  +  y^ixi  -  Xi). 

Notice  that  double  the  area  of  any  polygon  is  the  sum  of  the  products 
of  the  ordinate  of  each  vertex  by  the  difference  of  the  abscissas  of  the 
adjacent  vertices,  these  differences  being  taken  in  the  same  direction, 
anti-clockwise,  around  the  entire  polygon. 

13.  Find  the  area  of  the  triangle  whose  vertices  are  (12,  -  5),  (-  8,  7), 
(10,  15). 

14.  Find  the"  condition  that   (x,  y)  lie  in  the  straight  line  through 

(x',  2/').  (*"'  y")- 


RECTANGULAR   COORDINATES  7 

15.  Show  that  the  points  (1,  4),  (3,  2,),  (-  3,  8)  lie  in  a  straight  line. 

16.  The  vertices  of  a  pentagon  are  (-J,  3),  (—  5,  8),  (11,  —  4),  (0,  12), 
(14,  7).     Plot  the  pentagon  and  find  its  area. 

17.  A  piece  of  land  is  bounded  by  straight  lines.  From  the  survey 
the  rectangular  coordinates  of  the  stations  at  the  corners  referred  to 
a  N.  S.  line  and  an  E.  W.  line  through  station  A  are  as  follows,  distances 
measured  in  chains : 

^00  D     22.85     17.19 

B   14.30     -  15.04  E       7.42     40.09 

C  22.85       -4.18  F    -8.29     29.80 

Plot  the  survey  and  find  the  area  of  the  piece  of  land. 

18.  Find  tiie  point  which  divides  the  line  from  (x',  y')  to  (x",  y") 
internally  into  segments  whose  ratio  is  r. 

19.  Find  the  point  which  divides  the  line  from  (x',  y')  to  (x",  y") 
externally  into  segments  whose  ratio  is  r. 

20.  Locate  the  points  (2,  —  9),  (—  fi,  5),  and  also  the  points  dividing 
the  line  joining  them  internally  and  externally  in  the  ratio  2  :  3. 

21.  Show  that  the  points  (x,  y),  (x,  —  y)  are  symmetrical  with  respect 
to  the  X-axis. 

22.  Show  that  the  points  (x,  y),  (—  x,y)  are  symmetrical  with  respect 
to  the  r-axis. 

23.  Show  that  the  points  (x,  y),  (— x,  ~  y)  are  symmetrical  with 
respect  to  the  origin. 


Art.  5.  —  Distance  between  Two  Points 

The  distance  between  the  points  (x',  ?/'),  (x",  y")  is  the  hypote- 
nuse of  the  right  triangle 
whose  two  sides  about  the 
riglit    angle    are    («'  —  x") 
and  {y'  —  y").     Hence 

d  =  ^{x'  -x"y  +  (y'-y"f. 

Problems.  —  1.  Find  distance 
between  the  points  (4,  2) ,  (7,  5) ; 

(-3,6),  (4,-9);  (0,8),  (7,0);  F,,;.  r,. 

(15,  -17),  (8,2);   (-4,  -7),  (-12,  -19). 

2.  Derive  formula  for  distance  from  (x',  y')  to  the  origin. 

3.  Find  distance  from  origin  to  (5,  9) ;  (7,-4);  (12,-15);  (■ 


Y 

^ 

iKv 

') 

^ 

^ 

(X," 

i") 

A 

X 

9,  14). 


8  ANALYTIC  GEOMETRY 

4.  Find  the  lengths  of  the  sides  of  the  triangle  whose  vertices  are 
(-3,  -2),  (7,8),  (-5,0). 

5.  Tlie  vertices  of  a  triangle  are  (0,  0),  (4,  —5),  (—2,  8).  Find 
the  lengths  of  the  medians. 

6.  Find  the  distance  between  the  middle  points  of  tlie  diagonals  of 
the  quadrilateral  whose  vertices  are  (2,  3),  (—4,  5),  (6,  -  3),  (U,  7). 

7.  Show  that  the  points  (6,  0),  (1^,  15),  (-  3,  -  12),  (-  7^,  -  3) 
are  the  vertices  of  a  parallelogram. 

8.  Find  the  center  of  the  circle  circumscribing  the  triangle  whose  ver- 
tices are  (2,  2),  (7,  -  3),  (2,  -  8). 

9.  Find  the  equation  which  expresses  the  condition  that  the  point 
(x,  y)  is  equidistant  from  (4,  -  5),  (—  3,  7). 

10.  Find  the  equation  which  expresses  the  condition  that  the  distance 
from  the  point  (x,  y)  to  the  point  (  —  3,  2)  is  5. 

11.  Find  the  equation  which  locates  the  point  (x,  y)  in  the  circum- 
ference of  a  circle  whose  radius  is  r,  center  (a,  b). 

Akt.  6.  —  Systems  of  Points  in  the  1'lane 

If  any  two  quantities,  which  may  be  called  x  and  y,  are  so 
related  that  for  certain  values  of  x,  the  corresponding  values 
of  y  are  known,  the  different  pairs  of  corresponding  values  of 
X  and  y  may  be  represented  by  points  in  the  XF-plane. 

Comparative  statistics  and  experimental  results  can  fre- 
quently be  more  concisely  and  more  forcibly  presented  graphi- 
cally than  by  tabulating  numerical  values.  In  the  diagram 
the  abscissas  represent  the  years  from  1878  to  1891,  the  corre- 
sponding ordinates  of  the  full  and  dotted  lines  the  production 
of  steel  in  hundred  thousand  long  tons  in  the  United  States  and 
Great  Britain  respectively.*  The  diagram  exhibits  graphically 
the  information  contained  in  the  adjacent  table,  condensed 
from  "  Mineral  Kesources,"  1892.    Observe  that  if  the  points  are 

*  In  the  figure  the  linear  unit  on  the  X-axis  is  5  times  the  linear  unit 
on  the  r-axis.  It  will  be  noticed  that  the  essential  feature  of  a  system 
of  coordinates,  the  "one-to-one  correspondence"  of  the  symbol  (x,  y) 
and  the  points  of  the  A"l'-plane,  is  not  disturbed  by  using  different  scales 
for  oidiuates  and  abscissas. 


RECTANGULAR   COORDINATES 


9 


inaccurately  located  the  diagram  becomes  not  only  worthless, 
but  misleading. 

45i 


1878    '79 

'80     '81 

'82      '83 

'S-t 
Fm. 

'85      'SO     '87 
T. 

'88      '89 

'90      '91 

U.S. 

G.  B. 

U.S. 

n.  p.. 

1878 

7.3 

10.6 

1885 

17.1 

19.7 

1879 

9.3 

10.9 

1886  ■ 

25.6 

23.4 

1880 

12.5 

13.7 

1887 

.33.4 

31.5 

1881 

15.9 

18.0 

1888 

29.0 

34.0 

1882 

17.4 

21.9 

1889 

33.8 

36.7 

1883 

16.7 

20.9 

1890 

42.8 

36.8 

1884 

15.5 

18.5 

1801 

39.0 

32.5 

The  table  furnishes  a  number  of  discrete  points  which  in  the 
figure  are  connected  by  straight  lines  to  assist  the  eye. 

Problems.  —  Exhibit  graphically  the  information  contained  in  the  fol- 
lowins?  tables : 


1894 


Cost  of  steel 

rails  per 

long  ton 

in  Penn.sylvania 

mills  f 

rom  18(_ 

.     (Mineral  Resources.) 

1867    $166.00 

1874 

$04.25 

1881 

$61.13 

1888 

$29.83 

1868       158.50 

1875 

68.75 

1882 

48.50 

1889 

29.25 

1869       132.25 

1876 

59.25 

1883 

37.75 

1890 

31.75 

1870       106.75 

1877 

45.50 

1884 

30.75 

1891 

29.92 

1871       102.50 

1878 

42.25 

1885 

28.50 

1892 

30.00 

1872       112.00 

1879 

48.25 

1886 

34.50 

1893 

28.12 

1873       120.50 

1880 

67.50 

1887 

37.08 

1894 

24.00 

10 


ANALYTIC  GEOMETRY 


2.    Commercial  value  of  one  ounce  gold  in  ounces  silver  from  1855  to 
1894.     (Report  of  Director  of  Mint.) 


1855 

15.38 

1865 

15.44 

1875 

16.59 

1885 

19.41 

1850 

15.38 

1866 

15.43 

1876 

17.88 

1886 

20.74 

1857 

15.27 

1867 

15.57 

1877 

17.22 

1887 

21.13 

1858 

15.38 

1868 

15.59 

1878 

17.94 

1888 

21.99 

1859 

15.19 

1869 

15.60 

1879 

18.40 

1889 

22.09 

1860 

15.29 

1870 

15.57 

1880 

18.05 

1890 

19.76 

1861 

15.50 

1871 

15.57 

1881 

18.16 

1891 

20.92 

1862 

15.35 

1872 

15.63 

1882 

18.19 

1892 

23.72 

1863 

15.37 

1873 

15.92 

1883 

18.64 

1893 

26.49 

1864 

15.37 

1874 

16.17 

1884 

18.57 

1894 

32.56 

3.   Expense  of   moving  freight  per  ton  mile  on  N.Y.  C.  &  H.R.  R.R. 
from  1866  to  1894.     (Poor's  Railway  Manual.) 


1866 

<?'2.16 

1873 

J?  1.03 

1880 

^54 

1887 

fM 

1867 

1.95 

1874 

.98 

1881 

.56 

1888 

.59 

1808 

1.80 

1875 

.90 

1882 

.60 

1889 

.57 

1869 

1.40 

1876 

.71 

1883 

.68 

1890 

.54 

1870 

1.15 

1877 

.70 

1884 

.62 

1891 

.57 

1871 

1.01 

1878 

.60 

1885 

.54 

1892 

.54 

1872 

1.13 

1879 

.55 

1886 

.53 

1893 
1894 

.54 
.57 

4.   Pressure  of  saturated  steam  in  pounds  per  square  inch  at  intervals 
of  9°  from  32'^  to  428°  Fahrenheit.     (Based  on  Regnaulu's  results.) 


32° 

.085  lbs. 

131° 

2.27  lbs. 

230° 

20.80  lbs. 

329° 

101.9  lbs 

41 

.122 

140 

2.88 

239 

24.54 

338 

115.1 

50 

.173 

149 

3.02 

248 

28.83 

347 

129.8 

59 

.241 

158 

4.51 

257 

33.71 

356 

145.8 

68 

.333 

167 

5.58 

260 

39.25 

305 

163.3 

77 

.456 

176 

6.87 

275 

.45.49 

374 

182.4 

86 

.607 

185 

8.38 

284 

52.52 

383 

203.3 

95 

.800 

194 

10.16 

293 

00.40 

392 

225.9 

104 

1.06 

203 

12.20 

302 

09.21 

401 

250.3 

113 

1.38 

212 

14.70 

311 

79.03 

410 

276.9 

122 

1.78 

221 

17.53 

320 

89.80 

419 

428 

305.5 
336.3 

RECTA  NG  ULA  R   COOR  DIN  A  TES 


11 


In  these  problems  it  is  evident  that  theoretically  there  corresponds 
a  determinate  value  of  the  ordinate  to  every  value  of  the  abscissa.  Hence 
the  ordinate  is  called  a  function  of  the  abscissa,  even  though  it  may  be 
impossible  to  express  the  relation  between  ordinate  and  abscissa  by  a 
formula  or  analytic  function. 

5.  Suppose  a  body  falling  freely  under  gravity  down  a  vertical  guide 
wire  to  have  a  pencil  attached  in  such  a  manner  that  the  pencil  traces 
a  line  on  a  vertical  sheet  of  paper  moving 
horizontally  from  right  to  left  with  a  uni- 
form velocity.  To  determine  the  relation 
between  the  distance  the  body  falls  and  the 
time  of  falling.* 

Take  the  vertical  and  horizontal  lines 
through  the  starting  point  as  axes  of  refer- 
ence, and  let  01,  12,  23,  •••,  be  the  equal 
distances  through  which  the  sheet  of  paper 
moves  per  second,  the  spaces  05,  510,  •■•,  on 
the  vertical  axis  represent  5  feet.  Then 
the  ordinate  of  any  point  of  the  line  traced 
by  the  pencil  represents  the  distance  the 
body  has  fallen  during  the  time  represented 
by  the  abscissa  of  the  point.  Careful  meas- 
urements show  that  the  distance  varies  as 
the  square  of  the  time.  Calling  the  distance 
.s',  the  time  t,  the  distance  the  body  falls 
the  first  second  \  g,  where  g  is  found  by 
experiment  to  be  32.16  feet,  the  relation 
between  ordinate  and  abscissa  of  the  line 
traced  by  the   pencil   is   expressed   by  the 


-1 

r 

0 

1 

I  3 

4 

5 

\ 

10 

\ 

15 

^ 

. 

20 

\ 

25 

\ 

30 

\ 

35 

40 

45 

50 

55 

60 

65 

3 

proportion 


,  which  leads  to  the  equa- 


tion .<?  =  \  gt'.  The  curve  and  the  equation  express  the  same  physical 
law,  tJK!  one  algebraically,  the  other  geometrically. 

In  this  problem  the  ordinate  is  an  analytic  function  of  the  abscissa,  for 
the  relation  between  the  two  is  expressed  by  a  formula. 

The  ordinate  is  a  continuous  function  of  the  abscissa  ;  that  is,  the 
difference  between  two  ordinates  can  be  made  as  small  as  we  please  by 
sufficiently  diminishing  the  difference  between  the  corresponding 


*  This  is  the  principle  of  Morin's  apparatus  for  determining  cxperi 
mentally  the  law  of  falling  bodies. 


12  ANALYTIC  GEOMETRY 

6.  A  body  is  thrown  horizontally  with  a  velocity  of  v  feet  per  second. 
The  only  force  disturbing  the  motion  of  the  body  taken  into  account  is 
gravity.     Find  the  position  of  the  body  t  seconds  after  starting. 

Calling  the  starting  point  the  origin,  the  horizontal  and  vertical  lines 
through  the  origin  the  A'-axis  and  F-axis  respectively,  the  coordinates  of 
the  body  t  seconds  after  starting  are  x  —  vt,  y  =  —  ^  gfi.     Eliminating  t, 

y  = 2_  X-,  an  equation  which  expresses  the  relation  existing  between 

2  v^ 
the  coordinates  of  all  points  in  the  path  of  the  body. 


CHAPTER    II 


EQUATIONS   OF   GEOMETRIO   EIGUEES 
Art.  7.  —  The  Straight  Line 

A  point  moving  in  a  plane  generates  either  a  straight  line 
or  a  plane  curve.  Frequently  the  geometric  law  governing  tlie 
motion  of  the  point  can  be  directly  expressed  in  the  form  of 
an  equation  between  the  coordinates  of  the  point.  This  equa- 
tion is  called  the  equation  of  the  geometric  figure  generated 
by  the  point. 

Draw  a  straight  line  through  the  origin.     By  elementary 

geometry  -^"  =  -^  =  ^^  =  •  •  •.    This  succession  of  equal  ratios 

..Irt  Aa^  Aa.2 
expresses  a  geometric  property 
which  characterizes  points  in  the 
straight  line ;  for  every  point  in 
the  line  furnishes  one  of  these 
ratios,  and  no  point  not  in  the 
straight  line  furnishes  one  of  these 
ratios.  Calling  the  common  value 
of  these  ratios  m,  and  letting  x 
and  y  denote  the  coordinates  of 
any  point  in  the  line,  the  equation 
y  =  mx  expresses  the  same  geo- 
metric property  as  the  succession  of  equal  ratios.  Hence  if  tlie 
point  (x,  y)  is  governed  in  its  nurtion  by  the  equation,  it  generates 
a  straight  line  through  the  origin.  Uy  trigonometry  m  is  the 
tangent  of  the  angle  through  which  the  X-axis  must  be  turned 
anti-clockwise  to  bring  it  into  coincidence  with  the  straight  line. 
13 


Y 

/ 

/ 

7 

y 

r 

I 

>/ 

/ 

A 

/ 

X 

/ 

1   i"  T 

14 


ANALYTIC  GEOMETRY 


Y 

/^ 

/ 

/^ 

// 

/ 

^-' 

/ 

/^ 

y 

■7- 

n 

y 

X— 

A 

This  angle  is  called  the  angle  which  the  line  makes  with  the 
X-axis,  and  its  tangent  is  called  the  slope  of  the  line. 

Give  the  straight  line  y=^mx  a  motion  of  translation  parallel 
to  the  y-axis  upward  through  a  distance  n.  The  ordinate  of 
every  point  in  the  line  in  the  new  position  is  n  greater  than 
the  ordinate  of  the  same  point  in  the  line  through  the  origin. 

Hence  the  equation  of  the 
straight  line,  whose  slope  is  m, 
and  which  intersects  the  l''-axis 
at  a  point  n  linear  units  above 
the  origin,  is  2/  =  '"'-^  -f  ^-  ''^ 
is  called  the  intercept  of  the 
line  on  the  Y-axis,  x  and  y 
are  called  the  current  coordi- 
nates of  the  straight  line,  m 
and  n  are  called  the  parameters 
of  the  straight  line.  To  every 
straight  line  there  corresponds  one  pair  of  values  of  m  and  n ; 
for  a  straight  line  makes  only  one  angle  with  the  X-axis,  and 
intersects  the  T-axis  in  only  one  point;  conversely,  to  every 
pair  of  values  of  m  and  n  there  corresponds  only  one  straight 
line. 

Problems. —  1.  Write  the  equation  of  the  line  parallel  to  the  F-axis 
at  a  distance  of  5  linear  units  to  the  right  of  the  F-axis. 

2.  Write  the  equation  of  the  line  parallel  to  tlie  X-axis  intersecting 
the  F-axis  6  below  the  origin. 

3.  Write  the  equation  of  the  straight  line  through  the  origin  making  an 
angle  of  45°  with  the  A'-axis. 

4.  Find  the  equation  of  the  line  making  an  angle  of  135°  with  the 
X-axis,  intersecting  the  I'-axis  5  above  the  origin. 

5.  Write  the  equation  of  the  line  whose  slope  is  2,  intercept  on 
F-axis  —  5. 

6.  Find  the  equation  of  the  path  of  a  point  moving  in  such  a  manner 
that  it  is  always  equidistant  from  (3,  —  5),  (—  3,  5). 

7.  Find  the  equation  of  the  path  of  a  point  moving  in  such  a  manner 
that  it  is  always  equidistant  from  (4,  2),  (-  3,  5). 


EQUATIONS   OF  GEOMETRIC  FIGURES  15 

8.  Find  the  equation  of  the  locus  of  the  points  equidistant  from  (7,  4), 
(-3,  -5). 

9.  Find  the  equation  of  the  straight  line  bisecting  the  line  joining 
(2,  —  5),  (G,  3)  at  right  angles. 

AiiT.  8.  —  The  Circle 

According  to  the  geometric  definition  of  the  circle  the  point 
(x,  y)  describes  the  circumference  of  a  circle  with  radius  r, 
center  (a,  b),  if  the  point  (x,  y)  moves  in  the  XF-plane  in  such 
a  manner  that  its  distance  from  (a,  h)  is  always  r.  This  con- 
dition is  expressed  by  the  equation  {x  —  a)-  +  (y  —  b)-  =  r, 
which  is  therefore  the  equation  of  a  circle. 

Problems.  —  1.  Write  the  equation  of  the  circle  whose  radius  is  5, 
center  (2,  -  3). 

2.  Find  the  equation  of  the  circle  with  center  at  origin,  radius  r. 

3.  Find  equation  of  circle  radius  5,  center  (5,  0). 

4.  Find  equation  of  circle  radhis  5,  center  (5,  5). 

5.  Find  equation  of  circle  radius  5,  center  (—5,  5). 

6.  Find  equation  of  circle  radius  5,  center  (—5,  —  5). 

7.  Find  equation  of  circle  radius  5,  center  (0,  —  5). 

8.  Find  equation  of  circle  radius  5,  center  (0,  5). 

Art.  9.  —  Thk  Comc  Sections 

After  studying  the  straight  line  and  circle,  the  old  Greek 
mathematicians  turned  their  attention  to  a  new  class  of  curves 
which  they  called  conic  sections,  because  these  curves  Avere 
originally  obtained  by  intersecting  a  cone  by  a  plane.  Tt  was 
soon  discovered  that  these  curves  may  be  defined  thus : 

A  conic  section  is  a  curve  traced  by  a  point  muving  in  a 
plane  iiisucli_a  manner  tliatthe  ratio  of  the  distances  from  the 
moving  point  to  a  fixed  point  and  to  a  fixed  line  is  constant. 

This  definition  will  be  used  to  construct  these  curves,  to 
obtain  their  properties,  and  to  find  their  equations.  The  fixed 
point  is  called  the  focus,  the  fixed  line  the  directrix  of  the 
conic  section.  When  the  constant  ratio,  called  the  character- 
istic ratio  and  denoted  by  e,  is  less  than  unity,  the  curve  is 


16 


ANALYTIC  GEOMETRY 


called  au  ellipse ;    when  greater  than  unity,   an  hyperbola ; 

when  equal  to  unity,  a  parabola.* 

The  following  proposition  is  due  to  Quetelet  (1796-1874),  a 

Belgian  scholar : 

If  a  right  circular  cone  is  cut  by  a  plane,  and  two  spheres 

are  inscribed  in  the  cone  tangent  to  the  plane,  the  two  points 

of  contact  are  the  foci  of  the  section  of  the  cone  by  the  plane ; 

and  the  straight  lines  in  which  this  plane  is  cut  by  the  planes 

of  the  circles  of  contact  of  spheres  and  cone  are  the  directrices 

corresponding  to  these  two  foci  respectively. 

Let  the  plane  cut  all  the  elements  of  one  sheet  of  the  cone. 

F,  F'  are  the  points  of  contact  of  the  spheres  with  the  cutting 

plane;  F  any  point  in  the  intersection  of  plane  and  surface  of 
cone ;  T,  T'  the  points  of  contact 
of  element  of  cone  through  P 
with  spheres.  The  plane  of  the 
elements  Sa,  Sa'  is  perpendicular 
to  the  cutting  plane  and  the  plane 
of  the  circles  of  contact.  Since 
tangents  from  a  point  to  a  sphere 
are  equal,  PF=  FT,  FF  =  FT'. 
Hence 

FF  +  FF'  =  FT+  FT'  =  TT', 

Fig.  11.  a    constant.      Through    F    draw 

DD'  perpendicular  to  the  parallels  HH',  KK'.     From  the  simi- 


lar triangles  FDT  and  FD'T', 


FT 
FT 


PD 
FD' 


hence 


r>y  composition 


FF      PD 


TT 


-,  by  interchanging  means, 
1)1/   ^  *    *  '  FD 


PF  ^  PD 
FF'     PD'' 
PF  _  TV 
DD'' 


J^W"        PD'     P77" 

a  constant.     Similarly,  ^^-  =  -^,  i-^ 
^'TT'      DD''  PD' 


TT 
DD 


•    Call  the  points 


*  Cayley,  in  the  article  on  Analytic  Geometry  in  tlie  Britannioa,  niiiMi 
edition,  calls  this  definition  of  conic  sections  the  definition  of  Apolloiiius. 
ApoUonius,  a  Greek  mathematician,  about  203  b.c,  wrote  a  treatise  on 
Conic  Sections. 


EQUATIoys    OF  GKOMICTUHJ   FKiUUES 


17 


of  intersection  of  the  straight  line  FF'  with  the  section  of 
the  cone  V,  V'.     Since 

VF  +  VF  =  FF'  +  2  VF  =  TT 
and  VF  +  VF'  -  FF  +  2  VF  =  TT', 

VF=VF' 
and  T  'F  +  VF  =  VF'  +  1 7^"  =  I '  V  =  2'7^ 

Hence  the  constant  ratio  77^  =  7777,  i^^  l*^ss  than  unity,  and  the 

conic  section  is  an  ellipse. 


It  is  seen  that  the  ellipse  may  also  be  defined  as  the  locus  of 
the  points,  the  snni  of  whose  distances  from  two  fixed  points, 
the  foci,  is  constant. 

Let  the  plane  cut  both  sheets  of  the  cone.     With  the  same 

notation  as  before,  PF=  PT  PF' =  PT' ;  lienee 

PF-  PF'  =  TV  =  a  constant. 

,    PT      PD 
From   the    similar   triangles    PDT    and    PD'T ,   ^^,=  7777,; 

PP       PD  PF       PI)     ,  '  ^        ^  ^ 

hence   —  =  -— .     P.y  division  £±-  =  ±^-  hence 
PF'      PD'        ^  TV     DD' 


PF  ^  TV 
PD     DD' 


a  constant. 


18 


ANALYTIC  GEOMETRY 


Similarly,  = 


is  greater  than  unity,  and  the 


PD'     DD''     DB' 

conic  section  is  an  hyperbola. 

The  hyperbola  may  also  be  defined  as  the  locus  of  points,  the 
difference  of  whose  distances  from  the  foci  is  constant. 

Let  the  cutting  plane  and  the  element  MN  make  the  same 
angle  6  with  a  plane  perpendicular  to  the  axis  of  the  cone. 
The  intersections  of  planes  through  the  element  3fN  with  the 
cutting  plane  are  perpendicular  to  the  intersection  of  cutting 
plane  with  plane  of  circle  of  contact. 

FF=:  FT—  MN=  PD,  and  the  conic  section  is  a  parabola, 
focus  F,  directrix  HIl'. 

Art.  10. — The  Ellipse,  e<l 
Construction.  —  Let    F   be    the    focus,   HH'   the    directrix. 
Througli  F  draw  i^Vr  perpendicular  to  HH',  &nd  on  the  perpen- 
dicular to  FK  through 
F   take    the    points   P 
and  P'  such  that 
PF  ^P'F^ 
FK     FK     ^' 
Through  K  and  P,  and 
through  A"  and  P',  draw 
straight   lines.       Draw 
any  number  of  straight 
lines   parallel   to  HH', 
intersecting    KP     and 
KP'  in  r?„  n^,  n^,  v^,  •••, 
FK  in  ?)?j,  m-j,  vi^,  •••. 
With  F  as  center  and 
mj?;,  as  radius  describe 
an  arc  intersecting  niit, 
in  R.     Then 


H 

/ 

r' 

/ 

/ 

/ 

/ 

n 

/ 

/ 

r 

'i 

^ 

^ 

"'"' 

y 

-~^ 

^ 

T 

/ 

.' . 

' 

N 

K 

y 

/ 

VF 

/>1 

r, 

^ 

A 

[ 

'  \ 

v' 

K 

\ 

V. 

ni] 

/ 

- 

— 

,N 

■. 

/ 

\ 

^ 

X 

^ 

,> 

< 

', 

^ 

\ 

11 

\ 

\ 

\ 

\ 

\ 

H 

\. 

TTliK 


A'      FK       ' 


EQUATIONS   OF  GEOMETRIC  FIGURES  l!» 

and  Pi  is  a  point  in  tlie  ellipse.     Similarly,  an  infinite  number 
of  points  of  the  cnrve  may  be  located. 

Definitions.  —  The  perpendicular  through  the  focus  to  the 
directrix  is  called  the  axis  of  the  ellipse.  The  axis  intersects 
the  curve  in  the  points  V  and  V',  dividing  FK  internally  and 
externally  into  segments  whose  ratio  is  e.  The  points  V  and 
F'  are  called  the  vertices,  the  point  A  midway  between  V 
and  V,  the  center  of  the  ellipse.  The  finite  line  VV  is  the 
transverse  axis  or  major  diameter,  denoted  by  2a;  the  line 
PjPi  perpendicular  to  VV  at  A  and  limited  by  the  curve  is 
the  conjugate  axis  or  minor  diameter,  denoted  by  26;  the 
finite  line  PP'  is  called  the  parameter  of  the  ellipse,  denoted 
by  2 p.  The  lines  KP  and  KP'  are  called  focal  tangents. 
The  ratio  of  the  distance  from  the  focus  to  the  center  to  the 
semi-major  diameter  is  called  the  eccentricity  of  the  ellipse. 

Properties. — The  foci  F  and  F'  are  equidistant  from  the 
center  A.  By  the  definition  of  the  ellipse  VF  =  e  •  VK, 
V'F=  e  '  VK     Subtracting,  FF'  =  e-  VV.     Dividing  by  2, 

AF  =  e  •  AV      Hence    e  =  ^^—~,    that    is,    in   the   ellipse   the 

o 
eccentricity  equals  the  characteristic  ratio. 

FP 

By  definition ?  =  e,  and  by  construction 

^  AK  ^ 

FP,=  An, 

By  definiti(m  eccentricity  e  =  =^ 
^  ^  AV  a 

From  the  figure,     VF  =  AV—  AF  =  a  —  ae  =  a (1  —  e)  ; 

VF  =  A  V  +  vlP  =  a  +  ae  =  a(l  +  e). 

FPand  VF'  are  called  the  focal  distances. 


VT+  V'T  _VF+  V'F_  ^^ 

2                      2 

Hence  ^17r=^. 
e 

^  .  .,            AF      Va-- 
a  eccentricity  e  = = 

-b' 

20 


A  NA  L  YTIC  GEOMETli  Y 


15y    clefinitiun    ^  =  e,    hence     FA^="'^^~^^     ^^=e, 

hence   V'K^''-^^-^^- 
e 

From  the  fiLnire  i^/f  =  ^lA"  -  .li^  =  'i  -  ae  =  "'^^~^'^ ; 


F'A'=  .1 A  +  AF'  =  1^  +  «e  =  li(l±i^. 


By  cletiiiitiou 

--^=  e,  hence 
FK       ' 


p  —  ail  —  e-)—  a[  1 -^ —   =  a— =  — 

\  a-     J       a^      a 


Equation — Take   the    axis   of    the   ellipse   as  X-axis,   the 
perpendicular  to  the  axis  through  the  center  as  F-axis.     Let  P 

be  any  point  of  the  curve, 
its  coordinates  x  and  y. 
Tlie  problem  is  to  express 
the  definition  PF=e-  PH 
by  means  of  an  e(]uation 
between  x  and  y.  The 
definition  is  equivalent  to 
PF''=e^-PH\  which  is 
the  same  as 

pff  +  (AD  +  AFf 
=  e\AK+ADy, 

which  becomes       y'^  +  (x  +  aey  —  e^[  -  +  a;  ] , 

reducincf  to  —  +  — — ^ =  1. 

a'     a\l  -  e') 

Since  the  point  (o,  b)  is  in  the  curve,  a\l  —  e^)  =  6',  and  the 

equation  finally  becomes   -  -f-^=  1, 


Y 

(0,?J 

) 

p 

H 

— 

— 

^ 

^ 

■^- 

— 

— 

~Z^ 

■\ 

/ 

^ 

^ 

\ 

K 

^ 

(«,n 

\, 

F 

A 

D 

1 

X 

\ 

/ 

•^ 

^ 

H 

EQUATIONS   OF  GEOMETUKJ   FIGURES  21 

Summary. —  Collfrtiiig   tlic   results   of   the   preceding   pani- 

graphs,  the  fuudanieutal  properties  of  the  ellipse  --  -f  •  ,  =  1  are  : 

a-      b'^ 

Distance  from  focus  to  extremity  of  conjugate  diameter    a 

Distance  from  center  to  directrix  - 

e 
Distance  from  focus  to  center  ae 

Distance  from  focus  to  near  vertex 
Distance  from  focus  to  far  vertex 

Distance  from  directrix  to  near  focus 
Distance  from  directrix  to  far  focus 
Distance  from  directrix  to  near  vertex 
Distance  from  directrix  to  far  vertex 


Eccentricity 

Square  of  semi-conjugate  diameter 

Semi-parameter 

Art.  11.  —  The  Hyperbola,  e>l 

Construction.  —  Draw  FK  through  the  focus  F  perpendicular 
to  the  directrix  ////'.     On  the  perpendicular  to  FK  through  F 

take  the  i)oints  P  and  P'  such  that  -— -  =  — -  =  e.     Through  K 

^  FA      FK 

and  P,  and  through  K  and  P'  draw  straight  lines.  Draw  any 
number  of  parallels  to  ////',  and  on  these  parallels  locate  points 
of  the  curve  exactly  as  was  done  in  the  ellipse.  The  hyperbola 
consists  of  two  infinite  branches.  The  vertices  Fand  V  divide 
FK  internally  and  externally  into  segments  whose  ratio  is  e. 
The  construction  shows  that  the  parallels  to  ////'  l)etween  V 
and  V  do  not  contain  points  of  the  curve.  The  notation  is  the 
same  as  for  the  ellipse. 


a(l-e) 

a{\  +  e) 

a{\  -  e2) 

e 

a(l  -1-  e2) 

e 

a(l-e) 

e 

a{\+e) 

e 

e-  («'- 

h-^y^ 

e  — 

O 

62=«i(l    - 

-e^) 

p  =  a{\-e 

.)  =  ^ 
a 

22 


ANAL  VriC  GEOMETli  Y 


Properties.  —  From  the  definition  of  the  hyi)erbohi  VF—  e  •  VK, 
V'F^e  ■  V'K.    Adding  FF'=e-  VV]  dividing  by  2,  jLF^e-AV. 
eccentricity,  that  is  in  the  hyperbola  also  the 


Hence 


AF 


characteristic  ratio  equals  the  eccentricity. 


\ 

\ 

/ 

// 

\ 

\ 

\ 

H 

n. 

// 

\ 

\ 

^ 

'R 

\ 

\ 

/ 

% 

s 

N^ 

P. 

/; 

\ 

s 

\ 

/ 

1/ 

F' 

v' 

A 

k\ 

/ 

V. 

/'F 

vu, 

m 

/ 

/ 

\ 

\ 

/ 

/ 

/ 

\ 

/ 

/ 

p-^ 

\ 

/ 

/ 

\ 

s 

/ 

/ 

/ 

\ 

/ 

/ 

Hi 

\ 

From  the  figure 

VF=AF-AV=  a{e  -  1) ;   V'F=  AF -\- AV  =  a{e  +  1). 

By  definition 


YK==e,  hence  VK=  ''^"-^^■,    1'^  =  e,  hence  V'K= 
e  FA 


a(e  +  1) 


VK 


From  the  figure  AK ^  AV  —  F/i  =  a  —  a 
From  the  figure 


e  —  1  _  a^ 
e 


FK 


^AF-AK  =  ''^'-%F'K 


AK-{-AF  = 


aO-  +  1) 


By  definition  pU=  c,  hence  p  =  a{e-  —  1). 


Equation.  —  To  find  the  equation  of  the  hyperbola  take  the 
axis  of   the  curve  as   X-axis,  the   perpendicular  to  the.  axis 


EQrATIO.XS    OF  GEOMKiniC   FKJl'llKS  2o 

tlii'oai;li  llic  (•('iittn-  as  I'-axis.  Let  /'  he  any  ituiiit  of  the  curve, 
its  coordinates  x  and  //.  The  piubleiu  is  to  express  the  delini- 
tiou  FF=e-  PH  by  lueaus  ot  an  e(iuation  between  x  and  y. 


\ 

Y 

/ 

\ 

H 

/ 

\ 

/ 

kp 

\ 

/ 

''/ 

\ 

\ 

/ 

' 

1 

\ 

/ 

/ 

f' 

A 

K 

F 

/ 

|D 

X 

\ 

— 

/ 

\ 

_ 

— 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

/ 

H' 

\ 

\ 

The  definition  is  equivalent  to  PF'  —  e'  •  PII ,  Avliich  is  tlie 
same  as  PD' -it{AD  —  AFf  =  <i-{AD  —  AKf,  which  becomes 

y-  +  (.c  —  ac)'  —  e-(  x  —  -]  ,  reducing  to  "— -J- =  1. 

\        ej  '        tt-      a'-(e-  —  1) 

IMaciug  <i'{(''  —  1)=  b-,  the  ecjuation  takes  the  form  ^  —  -.,  =  1. 

Since  Ir  =  a-c^  —  a-,  it  is  seen  from  the  figure  that  an  arc 
described  from  the  vertex  as  a  center  witli  a  radius  e(iual  to 
distance  from  focus  to  center  intersects  the  F-axis  at  a  distance 
b  from  the  center.  2  6  is  called  the  conjugate  or  minor  diameter 
of  the  hyperbola. 

Summary.  —  Collecting  the  results  of   the   preceding  para- 
graphs, the  fundamental  properties  of  the  hyperbola 

^.2 


=  1  an 


24 


A iVM  L  Y TIC   GEOMETlt  Y 


Distance  from  vertex  to  extremity  uf  coiijujjate  diameter     ae 
Distance  from  fucus  to  center 
Distance  from  focus  to  near  vertex 
Distance  from  focus  to  far  vertex 

Distance  from  directrix  to  near  vertex 

Distance  from  directrix  to  far  vertex 

Distance  from  directrix  to  near  focus 

Distance  from  directrix  to  far  focus 

Eccentricity 

Square  of  semi-conjugate  diameter 

Semi-parameter 


ae 

a{t 

-1) 

ait 

+  1) 

a(e 

-1) 

e 

fl(e 

+  1) 

e 

a(e 

2-1) 

e 

a(e 

2+1) 

e 

e  = 

(a2  +  ^2)^ 

a 

62. 

=  rt2(e2_  1) 

P  = 

.,(e'-l)J-- 

AuT.  12. 


Y 

/)t2 

H 

/ 

n 

/ 

P= 

/ 

^> 

^ 

^ 

<' 

P 

/- 

/ 

/ 

'/ 

K 

/ 

V 

/f 

D 

\ 

\ 

%s 

X 

\ 

•• 

\ 

~ 

— 

P' 

\ 

^^ 

^  ^ 

^ 

\ 

\ 

^ 

\. 

— 

~K 

fix 

\ 

\ 

Pa 

^. 

PiiE  Parabola,  e  =  l 

Through  F,  the  foctis,  draw  FK 
perpendictilar  to  HH.\  the  direc- 
trix. On  the  perpendicular  to  FK 
through  F  lay  off  FP  =  FF  =  FK. 
Draw  the  focal  tangents  KP  and 
KP',  draw  a  series  of  parallels  to 
HH',  and  locate  points  of  the  pa- 
rabola as  in  the  case  of  ellipse  and 
hyperbola.  From  the  figure,  it  is 
seen  that  the  distance  from  vertex 
to  focus  is  ^  p,  distance  from  vertex 
to  directrix  is  ^j),  the  parameter 
being  2 j^- 

To  find  the  equation  of  the  pa- 
rabola, take  the  axis  of  the  curve 
as  X-axis,  the  perpendicular  to  the 
axis  at  the  vertex  as  F-axis.  Let 
J*(x,  ?/)  be  any  point  in  the  curve. 


EQUATIONS   OF  GEOMETIIW   FIGURES 


25 


The  problem  is  to  express  the  deliiiitioii  PF  =  I)K  by  means 
of  an  eqnation  between  x  and  y.  The  definition  may  bo  written 
1^'^  =  UK',  which  is  the  same  as  PD'  +  TTf  =  ( VI>  +  VKf, 
wliich  becomes 


if  +  i^c-^pf^ix  +  llif, 


re(bicing  to  y-  =  2  jkv- 

A  parabola  whose  focus  and  directrix  are  known  may  be 
generated   mechanically  as   in- 
dicated in  the  figure. 

Problems.  —  1.  Construct  the  el- 
lipse wliosc  pariviiK'ter  is  G,  eccentri- 
city 2. 

2.  Construct  tlie  hyperbola  whose 
panuueter  is  S,  eccentricity  J. 

3.  Construct  the  ellipse  whose 
diameters  are  10  and  8.  Find  tlie 
equation  of  the  ellipse,  its  eccentri- 
city, and  parameter. 

4.  Construct  the  hyperbola  whose  diameters  are  8  and  C>.     Find  Uw. 
e(iuation  of  the  liyperbola,  its  eccentricity,  and  parameter. 

5.  Construct  the  parabola  whose  parameter  is  12  and  find  its  e(iuation. 

6.  Find  the  equation  of  the  ellipse  whose  eccentricity  is   j^,  major 
diameter  10. 

7.  The  diameters  of  an  hyperbola  are  10  and  0.     Find  distances  from 
center  to  focus  and  directri.K. 

8.  The  distances  from  focus  to  vertices  of  an  hyperbola  are  10  and  2. 
Find  diameters. 


H 

-==: 

■-^ 

^ 

^ 

^ 

^ 

--, 

P^ 

>< 

k 

^ 

•-. 

/ 

/ 

.... 

/ 

/ 

K 

\ 

F 

X 

- 

\ 

\ 

X 

\, 

9.    The  parameter  of  a  parabola  is  1-2.     Find  distance  from  focus  to 
point  in  curve  who.se  abscissa  is  8. 

10.  Find  diameters  of  the  ellip.se  whose  parameter  is  10,  eccentricity  I. 

11.  In  an  ellipse,  the  distance  from  vertex  to  dinctrix  is  0,  eccentri- 
city J.     Find  diameters  and  construct  ellipse. 


26 


ANALYTIC   GEOMETRY 


12.    In  the  ellipse  —  +  ■'"  =  1  show  that  the  distances  from  the  foci  to 
a-      h- 
the  point  (x,  y)  are  r  =  a  -  ex,  r'  =  a  +  ex.     r  and  »•'  are  called  the  focal 
radii  of  the  point  (;c,  //).     The  sum  of  the  focal  radii  of  the  ellipse  is  con- 
stant and  equal  to  2  a. 

x" 


13.    In  the  hyperbola 


^-  =  1  show  that  the  focal  radii  of  the  point 
5- 

The  constant  difference  of  the  focal 


(x,  ?/)  are  r  —  ex  -  a,  r  =  ex  +  a, 
radii  of  the  hyperbola  is  2  u. 

14.  Find  the  equation  of  the  ellipse  directly  from  the  definition:  The 
ellipse  is  the  locus  of  the  points  the  sum  of  whose  distances  from  the  foci 
equals  2  a. 

Take  the  line  through  the  foci  as  A'-axis,  the  point  midway  between 
the  foci  as  origin.  When  the  point  (x,  y)  is  on  the  F-axis  its  distances 
from  F  and  F'  are  each  equal  to  a. 
Call  AF  —  AF'  =  c,  the  distance  of 
(x,  y)  when  on  the  T-axis  from  the 
origin  b.  Then  ci^  -  c-  =  b'\  The 
geometric  condition  PF  +  PF'  =  2  a 
is  expressed  by  the  equation 


- 

Y 

^ 

,P 

x?/) 

/ 

l/ 

,y 

^ 

'\ 

\ 

/ 

^ 

>■ 

^ 

l\ 

\ 

\ 

F' 

A 

D 

'   ) 

X 

\ 

/ 

^ 

_ 

^ 

VyH  (x-r)H  V//-+(x  +  0'-^  =  2  ff, 


—       which  reduces  to 


=  1. 


The  definition  used  in  this  problem 

suggests   a  very  simple   mechanical 

construction  of  the  ellipse  whose  foci 

and    major    diameter    are    known. 

Fasten  the  ends  of  an  inextonsible  string  of  constant  length  2  a  at  the 

foci  F  and  F' .     A  pencil  point  guided  in  the  plane  by  keeping  the  string 

stretched  traces  the  ellipse. 

15.  Find  the  equation  of  the  hyperbola  directly  from  the  definition : 
The  hyperbola  is  the  locus  of  the  points  the  difference  of  whose  distances 
from  the  foci  is  2  a. 

Take  the  line  through  tlie  foci  as  X-axis,  the  point  midway  between 
the  foci  as  origin.     Call  AF  =  AF'  =  c,  c^  -  a"  =  U^. 

The  condition  PF'  -  PF=2a  leads  to  the  equation 


Vy'^  +  (X  +  c)2 


which  reduces  to 


\/y"-  +  (a 
1. 


c)-^ 


FA^UATIONS    OF  dKOMFTIUC   FKiUIiES  27 


The  inechiuiical  fonstnictidu  of  tho  hyperbola  is  effected  as  indicated 
in  tlie  figure. 


"" 

\ 

^ 

\ 

>^^ 

;;^- 

\ 

5^ 

r 

\ 

^ 

•^ 

=-^ 

^^ 

X 

/ 

\ 

r^ 

f-^^ 

^ 

'" 

/, 

ifj) 

\ 

^ 

r 

< 

--' 

/ 

'/ 

^ 

r 

^ 

/ 

/ 

V 

t^ 

^ 

F'/ 

A 

\ 

F 

/ 

\ 

/ 

\ 

/ 

\ 

y 

\ 

y 

\ 

^ 

Fic.  21. 

16.  Two  pins  fixed  in  a  ruler  are  constrained  to  move  in  grooves  at 
right  anf,des  to  each  other.  Show  that  every  point  of  the  ruler  describes 
an  ellipse  whose  semi-diameters  are  the  distances  from  the  point  to  tho 
pins.     Tliis  device  is  called  an  elliptic  compass. 

Tlio  folldwint;-  statcMneiits  may  help  to  form  an  idea  of  flio 
importance  of  the  conic  sections  : 

The  planets  and  asteroids  move  in  ellipses  with  the  sun 
at  one  focus. 

The  eccentricity  of  the  earth's  orl)it  is  about  j.\^. 

The  eccentricity  of  the  moon's  elliplic  }iath  about  the  earth 
is  about  J^. 

Nearly  all  comets  move  in  parabolas  with  the  sun  at  the  focus. 

Tho  caljlo  of  a  suspension  bridLfe,  if  the  load  is  uniformly 
distributed  over  the  horizontal,  takes  the  form  of  a  ]iaral)ola. 

A  projectile,  iinless  projected  vei-tically,  moves  in  a  ])arabola, 
if  the  earth's  attraction  is  the  only  distinl)ing  forcu^  takcui  into 
account. 


CHAPTER    III 

PLOTTING   or   ALGEBEAIO   EQUATIONS 
AuT.  13.  —  General  Theory 

The  locus  of  the  points  (x,  y)  whose  coordinates  are  the  pairs 
of  real  values  of  x  and  y  satisfying  the  equation  f(;x,  y)  =  0  is 
called  the  graph  or  locus  of  the  equation. 

Constructing  the  graph  of  an  equation  is  called  plotting  the 
equation,  or  sketching  the  locus  of  the  equation. 

An  equation  f{x,  y)  =  0  is  an  algebraic  equation,  and  ?/  an 
algebraic  function  of  x,  when  only  the  operations  addition,  sub- 
traction, multiplication,  division,  involution,  and  evolution 
occur  in  the  equation,  and  each  of  these  only  a  finite  number 
of  times. 

When  the  equation  has  the  form  y  =f(x),  y  is  called  an 
explicit  function  of  x;  when  the  equation  has  the  form 
f(x,  ?/)  =0,  y  is  called  an  implicit  function  of  .a*. 

The  locus  represented  by  an  equation  f(x,  y)  —  0  depends 
on  the  relative  values  of  the  coefficients  of  the  equation.  For 
mf{x,  y)  =  0,  wliere  m  is  any  constant,  is  satisfied  by  all  the 
pairs  of  values  of  x  and  y  which  satisfy  f(x,  ?/)  =  0,  and  by  no 
others. 

If  the  graphs  of  two  equations  /i(.)-,  y)  =  0,  /.(.r,  y)  =0  an' 
constructed,  the  coordinates  of  the  points  of  intersection  of 
these  graphs  are  the  pairs  of  real  values  of  x  and  y  which 
satisfy  /i  {x,  y)  =  0  and  f,  {x,  y)—0  simultaneously. 

Occasionally  it  is  possible  to  obtain  the  geometric  definition 
of  a  locus  directly  from  its  equation,  and  then  construct  the 
locus  mechanically.  The  equation  x^  -^-y"^  =  25  is  at  once  seen 
28 


PLOTTING    OF  ALGEBRA W  EQUATIONS  20 

to  represent  a  circle  with  center  at  oritijin,  radius  5.  In  general 
it  is  necessary  to  locate  point  after  point  of  the  locus  by  assign- 
ing arbitrary  values  to  one  variable,  and  computing  the  corrc- 
si)onding  values  of  the  other  from  the  equation. 

Akt.  14.  —  Locus  OF  First  Degukk  Equation 

The  locus  of  the  general  first  degree  equation  Itotween  two 
variables  x  and  y,  Ax  -\-  B>/  +  C  =  0,  is  the  locus  of 

^        b"^     B 

Moving  the  locus  represented  by  this  equation  parallel  to  the 
I'-axis  upward  through  a  distance  -'^,  increases  each  ordinate 

fi  ^  ... 

l)y  — .  Hence  the  equation  of  the  locus  in  the  new  position  is 
?/  =  — —  .r,  which  represents  a  straight  line  through  the  origin, 

since  the  ordinate  is  proportional  to  the  abscissa.  The  equa- 
tion Ax  -\-  By  +  C  =  ()  therefore  represents  a  straight  line  whose 
slope  is  —  --,  and  whoso  intercept  on  the  F-axis  is  —  — .  The 
intercept  of  this  line  on  the  X-axis,  found  by  placing  y  equal 
to  zero  in  the  equation  and  solving  for  x,  is  — -• 

The  straight  line  represented  by  a  lirst  degre(>  c(pi:iii(>u  may 
be  constructed  by  determining  the  point  of  intersection  of  th(( 
line  with  the  F-axis  and  the  slope  of  the  line,  by  dcici'mining 
any  point  of  the  line  and  the  slope  of  tlie  line,  by  determining 
the  points  of  intersection  of  the  line  with  tlie  coordinate  axes, 
by  locating  any  two  points  of  the  line. 

Problems.  — Construct  by  the  different  methods  the  linos  repn^sentcd 
by  the  equations. 

1.  2x  +  32/  =  6.  3.    \x-\y  =  \.  5.   •%;(  =  1- 

2.  j/  =  x-5.  4.    ix-4?/  =  2.  6.   ^+^-1. 


30 


ANALYTIC  GEOMETRY 


7.  Show  that  ?/-  —  2  .r>j  —  8  x-  =  0  represents  two  straight  lines  through 
the  origin. 

8.  Show  that  a  homogeneous  equation  of  the  ?ith  degree  between  x  and 
y  represents  n  straight  lines  through  the  origin. 

9.  Construct  the   straight  line  ^  4-  ^  =  1   and  the  circle  x-  +  2/"  =  25 

and  compute  the  coordinates  of  the  points  of  intersection.     Verify  by 
measurement. 


AiiT.  15.  —  Stuaigiit  Ltxe  through  a  Point 


Tliroiigh  the  fixed  point  (Xf,. 
an  an.ufle  «  with  the  X-axis. 


yo)  draw  a  straight  line  making 
Let  (x,  y)  be  any  point  of  this 
line,    d   the   distance   of    (;r,  ?/) 
from  (.r,|,  ?/(,).     From  the  figure 
X  —  a'o  =  d  cos  a,  y  —  ?/o  --=  d  sin  a, 
whence  x  =  .r,,  4-  d>  cos  a, 
y  =  y^^  +  d  sin  a, 
an d  y  —  ?/„  =  tan  a  (x — .r„).     That 
is,  if  a  i)oint  (a-,  y)  is  governed 
in  its  motion  by  the  ecpiation 
Fro.  22.  1/  ~  1/0  =  tan  a  (x  —  x,|), 

it  generates  a  straight  line 
through  (a'o,  ?/o),  making  an  angle  a  with  the  X-axis,  and  the 
coordinates  of  the  point  in  this  line  at  a  distance  d  from  (.r,„  ?/„) 
are  x  =  Xo-\-  d  cos  a,  y  =  yQ-\-  d  sin  a.  The  distance,  d,  is  posi- 
tive when  measured  from  (xq,  y^)  in  the  direction  of  the  side  of 
the  angle  a  through  (x„,  7/,,) ;  negative  Avhen  measured  in  the 
opposite  direction. 


Y 

■^ 

d 

{X 

}!> 

(X 

,^ 

< 

^ 

^ 

— 

A 

X 

siraiglit  lino 
terms  of  the 


Problems.  —  1.  Express  the  coordinates  of  a  point  in  tl 
through  (2,  ".)  making  an  angle  of  30'  with  tlio  A'-axis  i 
distance  from  (2,  3)  to  the  point. 

2.  On  the  straight  line  through  (o,  —  2)  making  an  angle  of  00"  with 
the  X-axis,  find  the  coordinates  of  the  points  whose  distances  from 
(?.,  -  2)  are  10  and  -  10. 


riOTTINa    OF  ALaEBRAW   ^JQUAriONS 


31 


3.  Write  the  oiiuation  of  the  s(rai,i;ht  line  tlimu-li  (  -  2,  5)  and  making 
an  angle  of  45    with  the  A'-axis. 

4.  Write  the  equation  of  the  straight  line  thniugh  (4,  -  1)  wiiose 
slope  is  J. 

5.  Find  the  distances  from  the  point  (2,  ;5)  to  the  points  of  intersection 
of  the  line  through  this  point,  making  an  angle  of  30°  with  the  A'-axis 
and  the  circle  x^  +  y'^  =  25. 

The  coordinates  of  any  point  of  the  given  line  are  x  =  2  +  (?cos.OO\ 
y  =  3  +  d  sin  30°.  These  values  of  x  and  y  substituted  in  the  equation  of 
the  circle  x^  +  ?/  =  25  give  the  equation  d^  +  (4  cos  30°  +  6  sin  30°)rt  =  12, 
which  determines  the  values  of  d  for  the  points  of  intersection. 

AiJT.  16.  —  Taxcexts 

To  plot  a  numerical  algebraic  equation  involving;  two  vari- 
ables, put  it  into  the  form  y=f(x)  if  possible.  Comi)uto  the 
values  of  y  for  different  values  of  x,  and  locate  the  points  whose 
coordinates  are  the  pairs  of  corresponding  real  values  of  x  and 
?/.  Connect  the  successive  points  by  straight  lines,  and  observe 
the  form  towards  which  the  broken  line  tends,  as  the  nunilicr 
of  points  locaiod  is  indofinitoly  increased.  This  limit  of  tlu^ 
broken  line  is  tlic  locus  of  tlie  eqnatiim. 

ExAiNrPLK.  —  Plot  ?/-  +  a--  =  9. 
Here  y  =  ±  V'.)  —  x-,  a-  =  d 
.T  =  — 4  —3 

ji  =  ±  V^^  0 

0  +1 

±3         •  ±2V2 

or  extracting  the  roots 

r  =  - ;;         -  2 

,/ =     0    ±  2.2;;7    ±2. SI'S 

//  has  two  iMimerically  equal  values  for  each  value  of  x. 
Hence  the  locus  is  symmetrical  w^ith  respect  to  the  X-axis. 
For  a  like  reason  the  locus  is  symmetrical  with  respect  to  the 


:  V'.)  -  y-. 

—  2 

-1 

±  VT) 

±2V2 

-f  2          +3 
±  Vr.           0 

+  4 

10              -f  1 

js    ± ;;    ±  2.siis 

+  2 

±  2.2;m 

0 

32 


ANAL  YTIC  GEOMETR  Y 


Y 


I'-axis.  For  values  of  a- >  +  3  and  for  values  of  a;<  — 3, 
y  is  imaginary.  Hence  the  curve  lies  between  the  lines 
a;  =  +3,  a;  =  —  3.  The  curve  also  lies  between  the  lines  ?/=  +3, 
y=—o.  Locating  points  of  the 
locus  and  connecting  them  by 
straight  lines,  the  figure  formed 
apjj roaches  a  circle  more  and 
more  closely  as  the  number  of 
points  located  is  increased.  The 
form  of  the  equation  shows  at 
once  that  the  locus  is  a  circle 
whose  radius  is  3,  center  the 
origin. 

Through  a  point  (.t,,,  ?/„)  of  the 
circle  an  infinite  number  of 
straight  lines  may  be  drawn.  The  coordinates  of  any  point  of 
the  straight  line  y  —  ?/o  =  tan  a{x  —  a;,,)  through  (.?(„  ?/(,),  making 
an  angle  a  with  the  X-axis  are  .t  =  ccq  +  dcos,  a,  y  —  ?/„  +  f'  sin  n. 
The  point  (x,  y)  is  a  point  of  the  circle  x"^  -{-y^  =  9  wlien 

(x„  +  d  cos  ay  +  (,Vo  +  (I  sin  a)'  =  9, 
that  is,  when 

(1)  (.i-,2  +  7/,/  -  9)  +  2  (cos  a  •  a-o  +  sin  a  ■  y„)d  +  (1~  =  0. 
Equation  (1)  determines  two  values  of  d,  and  to  each  of  these 
values  of  d  there  corresponds  one  point  of  intersection  of  line 
and  circle.  Since  the  point 
(xo,  ?/o)  is  in  the  circle  x^  -i- y'^  =  9, 
the  first  term  of  equation  (I)  is 
zero,  hence  the  equation  has  two 
roots  equal  to  zero  when 

cos  n  •  .T„  +  sin  a  •  v/,,  =  0, 

that  is,  when  tan  «  = 


.Vo 


Y 

\ 

/^ 

..'■o 

.'/o) 

/ 

\ 

V 

^. 

A 

N 

k^ 

\ 

/ 

\ 

\. 

^ 

To 


d  =  0  there  corresponds  the  point 
(xo,  ?/„),  and  when  both  roots  of 


PLOTTING    OF  AIj;  hlUlA  KJ   ICijUATIONS  :;:j 

eqiialiuu  (1)  ;iic!  zero,  Hit-  two  jH»iiil„s  ol'  iiitL'rsi'clitiii  of  the 
straight  line  // —  y,,  =  t;ui  <«  (,f  —  .r„)  and  the  eircle  x'- +  j/- —  \) 
coincide  at  {j\„  //„),  and  the  line  is  the  tangent  to  the  circle 
(.?•„,  _?/„).  Hence  the  eqnation  of  the  tangent  to  the  circle  x--\-y-=*J 
at  the  point  {x„,  y^  is 

which  reduces  to  xxy  +  yy^  —  9. 

A  tangent  to  any  curve  is  defined  as  a  secant  having  two 
points  of  intersection  with  the  curve  coincident.*  \\y  the 
direction  of  the  curve  at  any  point  is  meant  the  direction  of 
the  tangent  to  the  curve  at  the  point. 

The  circle  x?  4-  ?/-  =  i)  at  the  point  (.r,,,  //,,)  makes,  with  the 

X-axis,  tan~'(  —  "  " ).  At  the  points  corresponding  to  .*;  =  U  the 
angles  are  tan-'( ^  )=  loS^'oT'  and  tan-'f -^^- I  —  41'' 2.';'. 

Problems.  — 1.    Sliuw    that     -"'-"  +  ••'"=1    is    tanj'ent    to   the    ellipse 
•|2  +  p  =  l  at(:r,„2,,). 

2.  Show  that ^^-"  -  y'-'^  =  1  is  tani^oiit  to  the  hyperbola 

orr      «2 

^2-ft^=l  at(.ro,  yo). 

3.  Show  that  yun  -  p{x  +  re)  is  tangent  to  the  parabola  if-  -  '2  px  at 
(■'•0,  2/0)  • 

Ai;t.  17.  —  I'oix'is  (JF  Discontinuity 

KXAMI'LK.  —  Plot  >f  =  ^^-^^^■ 

X  —  'J 

a;  =  -cc  •••-;■>   -4    -,S    -2-1     (»   +1    -f-lj   +:;   + -|  ...  -f  ^ 

y  =  4-  1  ...  4-i    +'j,    +1   +\     0   -t   -2    Tco  +4    +l>i-...  +1 

*  The  secant  definition  of  a  tangent  is  due  to  Descartes  and  Fermat. 

D 


34 


ANAL  YTIC   GEO  ME  TR  Y 


From  X  =  0  to  x-  =  +  2,  y  is  negative  and  iiu-reases  iiuleli- 
nitely  in  numerical  valne  as  x  approaches  2.  From  x  —  +  2 
to  x  —  +  cTj,  y  is  positive  and  diminishes  from   +  qk>    to  +  1. 

y  is  negative,  and  decreases 
numerically  from  —  i  to  0 
wliile  X  passes  from  0  to 
—  1.  y  is  positive  and  in- 
creases to  + 1  from  x  =  —1 
to  a;  :=  —  Oj .  The  curve 
meets  each  of  the  two 
straight  lines  x  =  2  and 
II  =  1  at  two  points  infi- 
nitely distant  from  the 
origin. 

The  point  corresponding 
to  X  =  2  is  a  point  of  dis- 
continnity    of    the    curve. 
J,',,;.  25.  I'^^Ji"   if    two   abscissas   are 

taken,  one  less  than  2, 
tlie  other  greater  than  two,  the  difference  between  the  corre- 
sponding ordinates  approaches  infinity  Avhen  the  difference 
between  the  abscissas  is  indefinitely  diminished,  while  the 
definition  of  continuity  requires  that  the  difference  between 
two  ordinates  may  be  made  less  than  any  assignable  quantity 
by  sufficiently  diminishing  the  difference  between  tlie  corre- 
sponding abscissas. 


Y 

\ 

~ 

\ 

\ 

\ 

^ 

^^ 

— 

— 

— 

A 

\ 

^ 

X 

\ 

~ 

- 

— 

\ 

\ 

— 

AkT.   18. ASYMI'TOTES 

Example.  —  Plot  /  —  x"-  =  4.     Here  //  —  ±  V.ir  +  4. 
a;  =  -co 4         -3         -2         -1  0+1         +2 

+  3        +4       ...+CO 
Z/  =  ±oo-..  ±4.47    ±3.r>l    ±2.83    ±2.24    ±2   ±2.24    ±2.83 

±3.61   ±4.47"-  ±  ^ 


PLOTTli\G    OF   ALUKIIRAIC   EQUATIONS 


1/  lias  two  iiiiiiici'ically  ('(pial  iH'al  values  with  opposiit;  si^i's 
fur  every  value  of  x.  Tlie  values  of  //  iuereuse  iudetiuitely 
in  numerical  value  as  x  in- 
creases indeHnilely  in  iniuieri- 
cal  value.  It  now  l)ec(Mnes 
important  to  determine  whether, 
as  was  the  case  in  Art.  17,  a 
strai.n'ht  line  can  be  drawn 
whieh  meets  the  eurve  in  two 
points  iniinitely  distant  from 
the  origin.  The  [xjints  of  in- 
tt'rseetion  of  the  straight  line 
//  —  iiix  -f  "■  iiiiil  the  locus  of 
y-  —  x-  =  4  are  found  by  making  F"^-  '-''• 

these  equations  sinudtaneous.  Eliminating  y,  there  results  the 
equation  in  x,  (?/r  —  1)  x'  +  2  mnx  +  n-  —  4  ==  0.  The  problem 
is  so  to  determine  m  and  n  that  this  equation  has  two  infinite 
roots.  An  equation  has  two  infinite  roots  when  the  coefficients 
of  the  two  highest  powers  of  the  unknown  quantity  are  zero.* 
Hence  y  =  mx-\-n  meets  y"^  —  x^  =  4:  at  two  points  infinitely 
distant  from  the  origin  when  nv  —  1  =  0,  2  mn  =  0,  whence 
7/1  =  ±  1,  n  =  0.  There  are,  therefore,  two  straight  lines  y  =  x 
and  ?/  =  —  X,  each  of  which  meets  the  locus  of  y-  —  x-  =  4  at 
two  points  infinitely  distant  from  the  origin.  These  lines  are 
called  asymptotes  to  the  curve. 


\ 

^ 

Y 

/ 

^ 

\ 

y 

r/  1 

\ 

V 

y 

/ 

\ 

/ 

\/ 

/ 

\ 

X 

/ 

\ 

^ 

\ 

\ 

^ 

L 

^ 

\ 

/^ 

L_ 

^\l 

Problem.  —  Show  that  y 


±-x  are  asymptotes  to  the  liyperbola 
X-     y^      , 


a- 


62 


*  Place    x  =  -    in    (1)    ax"  +  hx"^  +  cx""-^  +  •■■  +  h-x^  +  Ix  +  m  =  0. 

There  results  (2)  a  +  bz  +  cr^  +  ...  +  kz"  ^  +  Iz"  •  +  mz"  =  0.  Eiinatioii 
(2)  has  two  zero  roots  wlit'ii  a  =  0,  b  -  0.  Hence  L'(iualion  (1)  h;i,s  two 
infinite  roots  when  a  =  0,  b  =  0. 


36 


ANA  L  VTIC  GEOMETli  Y 


y  = 


AiiT.  19.  —  Maximum  and  Minimum  Okdinatks 

EXAMI'LK.  l^lut  //  =  X''  —  1  X  -\-  7. 

^-y,...   -4  -3  -2  -1   0  +1  +1\   +2  +3.--+^ 
..-29  +1  +13  +13  +7  +1  -i  +1  +13. ..+00 


For  .T  =  +  1,  _v  =  +  1 ;  fol- 
ic =+  2,  ?/  =  +  1 ;  for  a;  =  11 
y=  —  \.  Hence  between  x=l 
and  a:  —  2  the  curve  passes 
below  the  X-axis,  turns  and 
again  passes  above  the  X-axis. 
At  the  turning  point  the  ordi- 
nate has  a  niinimuni  value ; 
that  is,  a  value  less  than  the 
ordinates  of  the  points  of  the 
curve  just  before  reaching  and 
just  after  passing  the  turning 
point.  The  point  generating 
the  curve  moves  upward  from 
a;  =  0  to  .T  =  —  1,  but  some- 
where between  x  =—  1  and 
X  —  —  2  the  point  turns  and 
starts  moving  towards  the 
X-axis.  At  this  turning  point  the  ordinate  is  a  maximum; 
that  is,  greater  than  the  ordinates  of  the  points  next  the  turn- 
ing point  on  either  side. 

To  determine  the  exact  position  of  the  turning  points,  let  x' 
be  the  abscissa,  ?/'  the  ordinate  of  the  turning  point.  Let  h 
be  a  very  small  quantity,  y,  the  value  of  y  corresponding  to 
X  =  x'  ±  h.  Then  ?/i  -  ?/'  must  be  positive  when  y'  is  a  mini- 
mum, negative  when  >/'  is  a  maximum.     Now 

,/,  -y'  =  (3  x'  -  7)  (  ±  //)  +  .'5  -v  (  ±  J'Y  +  ( ±  /')'• 
h  may  be  taken  so  small  that  the  lowest  power  of  h  deter- 


Y 

^ 

r 

:    T 

t  \ 

\ 

\ 

I 

T 

1   r 

J^ 

f^ 

1 

A 

X 

Fig.  27. 


PLOTTING    OF  ALCKHnAW  Ei^UATlONti  87 

mines  the  si!j,ii  of  //,  —  //'.*  //,  —  //'  (-an  tlicrciore  have  the  same 
sign  for  ±  h  only  when  the  coefficient  of  the  lirst  power  of  h 
vanishes.  This  gives  3  x*^— 7  =  0,  whence  x  =  ±  V^.  x  =  +  V|, 
rendering  y^  —  y'  positive  for  ±  li,  corresponds  to  a  minimum 
ordinate;  a;  =  — V|,  rendering  //i  —  ,v' negative  for  ± //,  corre- 
sponds to  a  maximum  ordinate.  _ 

ANhen  x  =  +  V|,  .'/  =  7  -  V"  V^I  =  -  .2  ;  wlien  x  =  -  Vf, 
y^7  +i^V21  =  U.2.t 

The  values  of  a;  which  make  //  —  0  are  the  roots  of  the  ctpui- 
tion  ur'  —  7  .f  +  7  =  0.  These  values  of  x  aiv.  tlie  abscissas  of 
the  points  wliere  the  locus  of  y  =  x^  —  7x  +  7  intersects  the 
X-axis.  X'  —  7  .«  +  7  =  0,  therefore,  has  two  roots  between 
-|-  1  and  -f  'J,  and  a  negative  root  between  —  .">  and  —  4. 

AUT.    20.  I'olXTS    OK    IXFLECTIOX 

ExAMi'LE.  —  Plot  y  +  x-y  —  x  =  0.     Here  //  — 


X  ^-cc 3    -  2     -  1     0     +1     +2     +  3  ...  -f  X) 

If  (x,  y)  is  a  point  of  the  locus,  (—  x,—  y)  is  also  a  point  of 
the  locus.  Hence  the  origin  is  a  center  of  symmetry  of  the 
locus.  A  line  may  be  drawn  through  the  origin  intersecting 
the  curve  in  the  symmetrical  points  /-•  and  P'.     If  this  line  is 

*  Let  s  =  ah^  +  hh^  +  civ'  +  (IW'  +  •••  be  an  infinite  scries  with  finite 
coefficients,  and  let  li  be  greater  numerically  than  the  largest  of  the  co- 
efficients b,  c,  <1,  ••-.     Tlien  hh  +  eh-  +  dh^  +  •••  <  /t   —    and 

\  —  h 

s  =  h\a  +  hh  +  '•//■•!  -h  ,1h^ ...)  =  /(,3(rt  i  ^1),  when  A<h  —  • 

1  —  ft 

When  h  is  indefinitely  diminished,  h  — '- — diminishes  indefinitely.     Con- 
1  —  h 

sequently  A  becomes  less  than  the  finite  quantity  a,  and  s  has  the  sign 

of  a/i*. 

t  This  method  of  examining  fur  maxima  and  minima  was  invented  by 

Fermat  (1590-l(iG;3). 


38 


ANALYTIC  G EOMETR Y 


turned  about  xi  until  P  coincides  with  A,  F'  must  also  coincide 
with  A.  The  line  through  A  now  becomes  a  tangent  to  the 
curve,  but  this  tangent  intersects  the  curve.  From  the  figure 
it  is  seen  that  the  coincidence  of  three  points  of  intersection 


Y_^ 


at  the  point  of  tangency,  and  the  consequent  intersection  of  the 
curve  by  the  tangent,  is  caused  by  the  fact  that  at  the  origin 
the  curve  changes  from  concave  up  to  convex  up.  Such  a  point 
of  the  curve  is  called  a  point  of  inflection. 

To  find  the  analytic  condition  which  determines  a  point  of 
inflection,  let  (a-,,,  yo)  be  any  point  of  //  +  x'-i/  —  x  =  0.  The 
coordinates  of  any  point  on  a  line  through  (.i-,„  ?/„)  are 
X  —  Xq  +  d  cos  a,  !i  =  //„  +  d  sin  a.  The  points  of  intersection 
of  line  and  curve  correspond  to  the  values  of  d  satisfying 
the  equation 

O/o  +  -V^/o  —  -^'o)  +  (sin  «  —  cos  «  +  2  cos  a  ■  a;,?/,,  +  sin  a  •  .r,,^)  d 
-f  (cos^  a  •  ?/o^  +  2  sin  a  cos  a  •  x^y)  d-  +  cos"  a  sin  a  •  d'^  —  0. 

The  first  term  of  this  equation  vanishes  by  hypothesis,  and  if 
the  coefticients  of  d  and  d-  also  vanish,  the  straight  line  and 
curve  have  three  coincident  points  of  intersection  at  (Xf,,  ?/o)- 
The  simultaneous  vanishing  of  the  coefticients  of  d  and  d'^ 
requires  that  the  equations 

sin  a  —  cos  a  -\-  2  cos  a  •  aVi,'/o  +  sin  a  •  xj^  —  0 
and  cos-  «  •  %-  +  2  sin  a  cos  a  ■  .r,,  =  0 

determine  the  same  value  for  tan  «.     This  gives  the  equation 


l-2av/„. 


;  reducing  to  ^/^)  —  o  x^fi/^  +  2  .(\,  =  0,  which  to- 


PLOTTING    OF  ALGEURAKJ  EQUATIONS 


81) 


gcther  with  //„-  +  x^{y„  —  a;o  =  0  determines  the  three  puiuts  of 
iutiection  (0,  0),  (V3,  ^V3),  (- V3,  -  \Vo). 

Art.  21.  —  Diametukj  Mictuod  of  Tlotting  Equations 
ExAMi'LE.  — riot 
y-  —  2 xy  -\-'S.xr  —  Hj x  —  0. 

1 1  ere  y  —  x  ±  Vl(J  x  —  2  x'. 

Draw  the  strai.^•ht  line  y  =  x. 
Adding  to  and  subtracting  from 
the  ordinate  of  this  line,  corre- 
sponding to  any  abscissa  x,  the 
cpiantity  VlO  x  —  2  x^,  the  corre- 
sponding ordinates  of  the  re- 
qnired  locus  are  obtained. 

This  locus  intersects  the  line 
y  =  X  when  VlGx—2xr=0,  that 
is  when  x  =  0  and  x  =  8.  y  is 
real  only  for  values  of  x  from  0 
to  8.  The  curve  intersects  the 
X-axis    when   x  =  VlG  x  —  2  a?,  '''"^-  '"'*• 

that  is  Avhen  x=^.    Points  of  the  curve  are  located  by  the 

table, 

x  =  0  +1  +2  +3 

Vl6  x-2  x'  =  0  ±Vl4         ±2V6         ±V3() 

+  4  -1-5  +6  +7  +8 

±4V2  ±V30  ±2VG  ±VT4  0 


Y 

/ 

/- 

\ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

1 

/ 

/ 

/ 

/ 

A 

/ 

/ 

X 

/ 

/ 

\ 

/ 

V 

/ 

' 

AUT.    22. SuMMAliV    OK    FliOl'KUTlKS    OF    LoOI 

From  the  discussions  in  the  i)receding  articles,  the  following 
conclusions  are  ol)tained : 

1.   If  the  absolute  term  of  an  eijuation  is  zero,  the  origin  is 

a  point  of  the  hicus  of  tlu'  equation. 


40  ANALYTIC  GEOMETIIY 

2.  To  iind  wlu've  the  locus  of  an  equation  intersects  the 
X-axis,  pkice  y  =  0  in  the  equation  and  solve  for  x ;  to  find 
where  the  locus  intersects  the  F-axis,  place  x=  0  and  solve 
for  y. 

3.  The  abscissas  of  the  points  of  intersection  of  the  locus  of 
ij  =  f{x)  with  the  X-axis  are  the  real   roots  of   the  equation 

4.  If  the  equation  contains  only  even  powers  of  y,  the  locus 
is  symmetrical  with  respect  to  the  X-axis  ;  if  the  equation  con- 
tains only  even  powers  of  x,  the  locus  is  symmetrical  with 
respect  to  the  Y-axis.  The  origin  is  a  center  of  symmetry  of 
the  locus  when  (—  x,  —  y)  satisfies  the  equation,  because  {x,  y) 
does. 

5.  Tlie  points  of  intersection  of  the  straight  line 

2/  — ^u^  tan«(:«  —  a-o) 

with  the  locus  of  J\x,  ?/)=  0  are  the  points  {x,  y)  correspond- 
ing to  the  values  of  d  Avhich  are  the  roots  of  the  equation 
obtained  by  substituting  x  =  x^  +  d  cos  a,  y  —  y^-}-  d  sin  a  in 
f{x,  y)—  0.  The  number  of  points  of  intersection  is  equal  to 
the  degree  of  the  equation,  and  is  called  the  order  of  the  curve. 

6.  The  distances  from  any  point  {xq,  y^  to  the  points  of  in- 
tersection of  the  straight  line  y  —  ?/„  =  tan  a(x.  —  x^  Avith  the 
locus  of  fix,  y)—0  are  the  values  of  d  which  are  the  roots 
of  the  equation  obtained  by  substituting  x  =  .Vo  -f-  d  cos  a, 
y  —  ?/,,  +  d  sin  a  in  J\x,  y)  —  0. 

7.  The  tangent  to  /(.r,  y)  =  0  at  (x*,,,  ?/„)  in  the  locus  is 
found  by  substituting  x=  .»■„  +  d  cos«,  y  =  ?/„  +  d  sin  a  in 
/(.t,  _?/)=  0,  equating  to  zero  the  coefficient  of  the  first  power 
of  d,  and  solving  for  tan  a.     This  value  of  tan  u  makes 

y-y,  =  timu(x-  x^) 

the  equation  of  the  tangent  tof(x,  _?/)=  0  at  (xq,  y^). 

8.  If  the  curve  f(x,  y)  =  0  has  infinite  branches,  the  values 
of  'tn.  and  u  found  by  substituting  mx  +  n  for  //  in  the  equation 


PLOTTING    OF  ALGEIiUAW  EQUATIONS  41 

f{x,ij)=0,  and  tNiuating  to  zero  the  coetticients  of  tlio  two 
liighest  powers  of  x  in  the  resultini^  equation,  deteruiine  tlie 
line  //  =  mx  -\-  u  which  meets  the  curve  at  two  points  at  in- 
finity; that  is,  the  asymptote. 

9.  To  examine  the  locus  of  ?/=/(.»•)  for  maximum  and  mini- 
mum ordinates  form /(.r  ± //)  —  /(.!•).  Equate  to  zero  the  co- 
efficient of  the  first  power  of  h,  and  solve  for  x.  The  values  of 
X  which  make  the  coefficient  of  the  second  power  of  h  positive 
correspond  to  minimum,  those  which  make  this  coefficient 
negative  correspond  to  maximum  ordinates. 

10.  To  determine  the  points  of  inflection  of  f(x,  y)=  0,  sul)- 
stitute  X  —  .To  -f  d  cos  a,  y  =  yQ-\-d  sin  cc  in  f(x,  y)  =  0.  In  the 
resulting  equation  place  the  coefficients  of  d  and  d'-  ecpial  to 
zero,  and  equate  the  values  of  tan  a  obtained  from  these  equa- 
tions. The  resulting  equation,  together  with  the  equation  of 
the  curve,  determines  the  points  of  inflection. 

Problems.  —  I'lot  tlie  numerical  algebraic  equations: 

1.    2x  +  :],j~7.  5.    (X- 2)0/ +  2)=  7.    10.    yi  =  -\Ox. 

2  ^^U=\  6-  ■'■^  +  y''  =  -^-  11-    ^  ■'■'-  +  ^>J^  =  36. 

'48'  7.    a:-2-?/2  =  25.  12.   4x^-9>/  =  m. 

3-  a-//  =  4.  8.    X-  -  ?/2  =  -  25.  13.   4  x-  -  0  ?/2  =  -  30. 

4.  (x-2)ij  =  [,.  9.    if=z\Ox.  14.    ?/-i  =  10.C-X2. 

15.  2/-  =  X-  -  10  r.  26.    ?/  =  x-  -  4  .c  +  4. 

16.  x^  +  10  ./•//  +  >/i  ^  25.  27.    2/2  =  (a:  +  2)  (X  -  3). 

17.  .t2  +  10  x)/  +  >/  +  25  =  0.  28.    2/2  =  x2  -  2  x  -  8. 

18.  2/"^  =  8  x2  -  x^  +  7.  29.   2/"  =  x2  -  4  x  +  4. 

19.  x2  +  2  xy  +  y2  ^  25.  30.    2/  =  (x  -  1  )(x  -  2) (x  -  3) . 

20.  .r2  +  10 .r.v  +  f  =  0.  31.    y^  =  x'^  -  G  x"- +  \\  x  -  G. 

21.  I/-  =  .rt  -  x2.  32.   y  =  x*  -  5  x:-  +  4. 

22.  2/2  =  x2  -  X*.  33.    2/-  =  x«  -  5  x2  +  4. 

23.  2/-  =  x^  -  x\  34.    y  =  x»  -I-  2x3  -  3x2  -  4  X  -I-  4. 

24.  2/  =  (x  +  2)(x-3). 


35.    y=_^L_ 
2/  =  x2-2x-8.  -^      l-x'J 


42 


ANALYTIC  GEOMETRY 


2x-l 
3  X  +  5' 

y  +  5 
3  -  x' 

4-3x 
5x  —  6 

39.  ?/  -  2  x?/  -  2  =  0. 

40.  y-  +  2  x^  +  3  x"^  -  4  X  =  0. 

41.  m2  =  x3-2x-^-8x. 


36.  y 

37.  y 

38.  2/ 


42.  ?/  =  x3-9x2  + 24x  +  3. 

43.  ?/=(3x-5)(2x  +  9). 

44.  2/'i  =  x3  -  2  x2. 

45.  2/2 +  2x2/- 3x2  +  4x  =  0. 

46.  2/^ -2x2/ +  x2  +  X  =  0. 

47.  2/^  +  4  X2/  +  4  x2  -  4  =  0. 

48.  2/"-^  -  2  X2/  +  2  X'-  -  2  X  =  0. 

49.  2/--2x2/  +  2x2+22/  +  x  +  3  =  0. 

50.  2/"  -  2  X2/  +  x-  -  4  2/  +  X  +  4  =  0. 


1. 

x2  +  2x-  15  =  0. 

5. 

x^  -  7  X  +  7  =  0. 

2. 

x3-3x-  10  =  0. 

6. 

x'^  -  7  X  -  7  =  0. 

3. 

x2-4x  +  4  =  0. 

7. 

x3  +  7  X  +  7  =  0. 

4. 

x2  -  5  X  +  9  =  0. 

8. 

x3-5x  +  2  =  0. 

riot  the  real  roots  of  the  following  equations  : 

9.  X*  -  5  x'-  +  4  =  0. 

10.  xHa^'Hx-+x+l=0. 

11.  x^  +  X-  +  X  +  1  =  0. 

12.  x^  -  X-  +  X  +  1  =  0. 

riot  the  real  roots  of  the  following  pairs  of  simultaneous  equations  : 

1.  2/^  =  10x,  x^  +  2/2  =  25.  Plot  the.se  equations  to  the  same  axes. 
The  coordinates  of  the  points  of  intersection  of  the  loci  are  the  pairs  of 
real  values  of  x  and  y  which  satisfy 
each  of  the  given  equations.  The 
points  of  intersection  are  (2.07,4.42), 
(2.07,  -4.42). 

By  the  angle  of  intersection  be- 
tween two  curves  is  meant  the  angle 
between  the  tangents  to  the  curves  at 
their  intersection.  Hence  the  angle 
between  two  curves  is  the  differ- 
ence between  the  angles  the  tangents 
to  the  curves  at  their  intersection 
make  with  the  X-axis.  Calling  the 
angles  the  tangents  to  2/^  =  10  x  and 
x2  +  2/2  =  25  at  the  point  of  intersec- 
tion (xo,  2/0)  make  with  tlie  A'-axis 
a'  and  a  respectively, 

tan  a' 


-y 

^"        v^ 

z         /  \ 

y         7       V 

t          t       \ 

A                             X 

V       ^t          ^ 

\            \         L 

^           X   z 

^-       -< 

^\ 

PLOTTING   OF  ALGEllUAIC  EQUATIONS 


43 


Evaluating  for  a-o  =  2.07,  |/n  =  4.48,  tan  a'  =  1.10,  tan  a  =  —  .47  ;   whence 
a'  =  48°  4',  a  =  154°  50',  ami  the  anglo  between  the  curves  is  104°  52'. 

2.  2  X  +  3  y  =  5,  ?/  =  i  .'c  +  3.  8.  /-  +  ;/-  =  25,  ij-  =  10  x  -  x^. 

3.  y  =  3  X  +  5,  .T^  +  !/-^  =  25.  9.  3 .c^  +  2  y-^  =  7,  ?/  -  2  x  =  0. 

4.  X-  +  J/'-  =  9,  */2  =  10  X  —  x^.  10.  y-  =  4  X,  2/  —  X  =  0. 

5.  2/'^  =  10 X,  4  x2  -  0  y^  =  ;](!.  11.  2  x^  -  r'  =  14,  x^  +  2/^  =  4. 

6.  2  x-  —  IJ-  =  14,  X-  +  y'^  =  !).  12.  x-  -f  ?/-  =  25,  x'^  —  )/-  =  4. 

7.  >/  =  x^  —  7  X  +  7,  ?/  —  X  =  0. 


Solve  the  following  equations  graphically 


1.  x2  —  X  -  (5  =  0.  Plot  //  =  X-  and  y  =  x  +  C>  to  the  same  axes.  For 
the  points  of  intersection  of  the  loci  x-  =  x  +  d  ;  that  is,  x-  —  x  —  C  =  0. 
Hence  the  abscissas  of  the  jioints  of 
intersection  of  y  =  x^  and  y  —  x  +  0 
are  the  real  roots  of  x-  —  x  —  0  =  0. 
For  all  quadratic  equations, 

X-  +  ax  +  h  =  0, 

the  curve  ;/  =  x-  is  the  same,  and  the 
roots  are  the  abscissas  of  the  points 
of  intersection  of  the  straight  line 
y  =  —  ax  —  h  with  this  curve. 

In  like  manner  the  real  roots  of 
any  trinomial  equation  x"  +  rtx4-/;=0 
ai-e  the  abscissas  of  the  points  of  in- 
tersection of  y=x"  and  y  +  ax-\-b=0. 

2.  x--3x  +  2  =  0. 


3.  x"-  +  5  X  +  0  -- 

4.  .r2  -4=0. 

5.  :,•;:_  Ox -10: 


0. 


0. 


\ 

Y 

1 

\ 

I/. 

— 

I 

A 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

/ 

/ 

\ 

/ 

/ 

\ 

/ 

/ 

\ 

/ 

\ 

/ 

X 

A 

Fk! 

.•?!. 

6.  .r2-4x-  15  =  0. 

7.  3x2-12x  +  2=0. 

8.  x'-i+5x  +  10  =  0. 

9.  X--  5x  +  5  =  0. 


10. 

x3  -  7  X  +  7  =  0. 

11. 

x^  +  7  X  +  7  =  0. 

12. 

x^  +  7  X  -  7  =  0. 

13. 

.x«-7x-7  =  0. 

14. 

x'»-10x+  15  =  0. 

15. 

a:»-10x-15  =  0. 

44 


ANALYTIC  GEOMETRY 


Sketch  the  following  literal  algebraic  ecjuations : 

?/2  =  a;3 


1.  ?/2  _  3.3  _  (5  _  c)x^  —  bcx.  Here  y  —  ±  vx(x  —  6)  (x  +  c).  Unless 
numerical  values  are  assigned  to  b  and  c,  it  is  impossible  to  plot  the  equa- 
tion by  the  location  of  points.  How- 
ever, the  general  nature  of  the  locus  may 
be  determined  by  discussing  the  equa- 
tion. The  A'-axis  is  an  axis  of  sym- 
metry, the  origin  a  point  of  the  locus. 
For  0  <  X  <  6,  y  ifi  imaginary  ;  when 
X  =  b,  y  =  0 ;  for  x  >  b,  y  has  two  nu- 
merically equal  real  values  with  op- 
posite signs,  increasing  indefinitely  in 
numerical  value  with  x.  For  0  >  x  >  -  r, 
y  has  two  numerically  equal  values  with 
opposite  signs  ;  for  x  =  —  c,  y  =  0  ;  for 
^  <  —  <"»  2/  is  imaginary.  Sketching  a 
curve  in  accordance  witli  these  condi- 
tions, a  locus  of  the  nature  shown  in  the 
figure  is  obtained. 


ili 


=  1. 


X2 


Pig.  32. 
=  1. 


3.  ?/  =  ax.  6. 

4.  ?/  =  ax  +  b.  7. 


»/-  =  -J,  px. 

X-  +  y-  —  a-. 


b-^ 


b-^ 


=  1. 


13. 

(x-rt)(2/-  b)  =  m. 

14. 

y  =  (x-a)(x-b)ix-c). 

15. 

2/2  =  (x-a)(x-6)(x-0. 

16. 

V2-(X      a)-^^-^ 

17.  y-x  =4  a- (2  a  -  x). 

18.  ?/-  =  (x  -  a)  (x  -I-  /;)  (x  -  c). 


11.  ?/  =(x  -  a)(x  -f  ?;). 

12.  ?/- =(x- ffl)(x-(- ft). 

19.  ^y-=x^,  the  semi-cubic  parabola.* 

20.  rr//  =  x^,  the  cubic  parabola. 
?/'-  =  (x  —  f'i)(x  —  P2)(x  —  Cs),  ei,  real,  Ci,  cs,  conjugate  imaginarios. 
2/-  =  (x  -  Pi)  (x  -  eo)  (x  -  es),  <'i,  ^1;,  '';!,  real,  ^  >  r.2  >  ^3- 
y'  =  (x  —  (?i)(x  —  <'2)(x  -  P3),  fii,  ^25  Cti  real,  C]  =  r.:>,  ei  >  pj. 
?/-  =  (x  -  ei)(x  —  r2)(x  -  Pz),  P\,  e-2,  Cs,  real,  ^1  >  ^2,  ^2  =  ea- 
?/2  =:  (X  -  Pi)  (x  -  r.,)  (x  -  fs),  Ci  =  e-2  =  es- 


21 


25 


*  The  rifling  of  a  cannon,  wlion  the  bore  is  rolled  out  on  a  plane, 
technically  "  (Icvclopeil,"  is  a  srmi-cul)ic  paralmla. 


CHAPTER   IV 

PLOTTING  OF  TRANSCENDENTAL  EQUATIONS 
Art.  23.  —  Elemkntauv  Tiianscendkntal   Fltnctions 

Transoendental  e(]u<ati()iis  are  e(iuat.it»ns  involvinj;-  ti'aiiscon- 
dental  functions. 

The  elementary  transcendental  functions  are  the  exiioiion- 
tial,  logarithmic,  circular  or  trigonometric,  and  inverse  circular 
functions. 

The  expression  of  ti-anscendental  Cnnctioiis  by  means  of  tlu' 
fundamental  operations  of  algehra  is  possil)k^  only  hy  means 
of  infinite  series. 

AliT.    24. ExroXKXTIAL    and    LoOAUrTHMIC    FirXCTIONS 

The  general  type  of  the  exponential  function  is  y  =  h-a", 
where  a  is  called  the  base  of  the  exponential  function  and  is 
always  positive. 

To  plot  the  exponential  func- 
tion numerically,  suppose  />  =  1 , 
c  =  1,   ((  =  U.      Then   y  =  2''    and 

a;=  —  CO 4     — 3     — 2     — 1 

11=       0...     i,         I.         \         I 

0  1     2     3       4...r>D. 

1  2     4     S     1  ('....  X. 
Vov  all  values  of  a  the  locus 

of  y  =  a'  contains  the  point  (0,  J) 
and  indefinitely  approaches  the 
X-axis.  Increasing  the  value  of 
a  causes  the  locus  to  recede  more 
45 


_  Y     _ 

I        ,         ^  — 


46  ANALYTIC   GEOMETRY 

rapidly  from  the  X-axis  for  x  >  0,  and  to  approach  the  X-axis 
more  rapidly  for  x<().  When  «  =  !,  the  locus  is  a  straight 
line  parallel  to  the  X-axis.  When  a  <  1,  the  locus  approaches 
the  X-axis  for  x>l,  and  recedes  from  the  X-axis  for  x  <  0. 

When  c  is  not  unity  the  function  y  =  a"  may  be  Avritten 
y  =(cfy,  and  the  base  of  the  exponential  function  becomes  a". 
When  b  differs  from  unity,  each  ordinate  of  y  =  h  -  a"  is  the 
corresponding  ordinate  of  ?/  =  a"  multiplied  by  b. 

To  plot  the  exponential  function  y  =  b  •  a"  graphically,  com- 
pute ?/n  and  ?/i ,  the  values  of   y  corresponding  to  x  =  0  and 
X  =  a'l,  where  x^  is  any  number  not  zero.     Adopt  the  following 
notation  for  corresponding  values  of  x  and  y. 
x= 4a-,    -'Sxi     -2x,     -x,     0     x,     2x,     ?.x,     4.x,--- 

y  =  —     2/- 4       Vz        y-2     y-i  ih    Vx      y-2      y-s      yi---- 

Then  •lsl  =  l^  =  '!h^'!h  =  yi=.h=...  ««,.      On   two    intersect- 

y  2    y-i    2/o    yi    v-i    .Vs 

ing  straight   lines   take    OA  —  yo,   OB  =  yi.     Join  .1  and  B, 


B 


x./ \A. 


O         K      A  C  E  C. 

Fin.  .'54. 

draw  BC  making  angle  OBC  =  angle  OAB.     Then  draw  CD, 
DE,  EF,  •••,  parallel  to  AB  and  BC  alternately  ;  AH,  II K,  KL, 
•  ••,  parallel  to  BC  and  AB  alternately.     From  similar  triangles 
0K_  OII^  OA  ^OB^OC^OD^  OE 
()L~ 0K~  OII~  OA      OB      OC      01) 
Hence,  if    0.4  =  ?/o,   OB  =  yi,  it   follows  that  OL  =  y_:„  OK 


PLOTTING   OF  TIlANSCENDENTAL    EQUATIONS       47 

=  V  .,.  on  =  i/^x,  OC  —  y.2,  OD  =  II .^ ;  that  is,  the  ordi nates  cor- 
resi)onding  iox  =  —  ox^,  —  2:Ci,  —  x^,  2a-,,  Sifj  become  known 
and  the  points  of  the  curve  can  be  located. 

The  logarithmic  function  ex  =  log„  {by)  is  equivalent  to  tlie 
exponential  function  ?/  =     a".     When  y  =  log  a.-  is  plotted,  tlie 

logarithm  of  the  product  of  any  two  numbers  is  the  sum  of 
the  ordinates  of  the  abscissas  which  represent  the  nund)ers, 
and  the  product  itself  is  the  aV)Scissa  corresponding  to  this 
sum  of  the  ordinates. 

The  slide  rule  is  based  on  this  principle.  In  the  slide  rule 
the  ordinates  of  the  logarithmic  curve  are  laid  off  on  a  straight 
line  from  a  common  point  and  the  ends  marked  by  the  corre- 
sponding abscissas. 

Art.  25. — Circular  and  Inverse  Circular  Functions 


Ty\  definition,  am  AOP 


PD 
OP 


P'D' 
OP' 


angle   ^lOP    in   circular    measure    is 

Hence,  if  the  radius  OA'  is  the  linear 
unit,  the  line  P'D'  is  a  geometric  rep- 
resentative of  sinvlOP,  the  arc  A'P' 
a  geometric  representative  of  the 
angle  AOP.  The  measure  of  the 
angle  ylOPis  1  when  arc  AP=  OP; 
that  is,  the  unit  of  circular  measure 
is  the  angle  at  the  center  which  in- 
tercej^ts  on  the  circumference  an  arc 
equal  to  the  radius.  The  unit  of 
circular  measure  is  callcil  the  radi 
of    four   riglit    angles,   or   .".r.0°,  is  ' 


and  the  value 
arc  A  P     ai 


of  the 
c.I'P' 


OP 


OP 


radian  is  ecpiivalent  to 


360'' 


r°.3  -. 


48 


ANALYTIC   GEOMETRY 


Calling  angles  generated  by  the  anti-clockwise  motion  of 
OA  positive,  angles  generated  by  the  clockwise  motion  of  OA 
negative,  there  corresponds  to  every  value  of  the  abstract  num- 
ber X  a  determinate  angle. 

Unless  otherwise  specified,  angles  are  expressed  in  circular 
measure.  When  an  arc  is  spoken  of  without  qualiiieation,  an 
arc  to  radius  unity  is  always  understood. 

In  tables  of  trigonometric  functions  angles  are  generally  ex- 
pressed in  degrees.  Hence,  to  plot  y  =  sin  x  numerically, 
assign  arbitrary  values  to  x,  find  the  valne  of  the  correspond- 
ing angle  in  degrees,  and  take  from  the  tables  the  numerical 
value  of  sin  x. 


Y 

\ 

/ 

X 

\ 

\ 

^    ^^ 

/ 

A 

\^ 

-  200° 32' 

?/=  ••• 

+  .350 

3. 

-  ,S5°  57 

-  .997 

\ 

1 

28°  39' 

57°  18' 

.479 

.841 

3 

171°  53' 

.141 

-4 

-  143°  14' 

-  .598 

-114°  35 
-  .909 

-1 

-57°  18' 
-  .841 

-28°  39' 
-.479 

0 
0 
0 

85° 57'        114° 35'       143° 14' 
.997  .909  .598 


3  .... 


171' 


.141 


In  itvactical  problems  the  ecpiation  frequently  occurs  in  the 
form  y  =  sine  (ttx).     Were 


PLOTTING    OF   TRANSCENDENTAL    EQUATIONS       4'J 


?/=       0      Wli  1       W'2  0       -^V2       -1 

0   jV2   1   jV2   0    -|V2    -1    -^-V2 


^Vii 


Y 

lAtbz:: 


To  plot  ^  =  sin  X  grapliically,  draw  a  circle  witli  radius  unity, 
divide  the  circiunfereuce  into  any  number  of  equal  parts,  and 
placing  the  origin  of  arcs  at  the  origin  of  coordinates,  roll  the 
circle  along  the  X-axis,  marking  on  the  A'-axis  the  points  of 


division  of  the  circumference  0,  1,  2,  .'5,  4,  5,  (>,  •••.  Througli 
the  points  of  division  of  the  circumference  draw  perpendicu- 
lars to  the  diameter  through  the  origin  of  arcs  00,  11,  22,  «*>.'», 
44,  55,  GG,  •■•.  On  the  perpendiculars  to  the  X-axis  at  the 
points  0,  1,  2,  o,  4,  5,  G,  •••,  lay  off  the  distances  00,  11,  22,  ',y,\, 
44,  55,  GG,  •••,  respectivt'ly.  lu  this  inauucr  any  nundjer  of 
points  of  y  =  sin  x  may  be  located. 


50 


ANALYTIC  GEOMETRY 


On  account  of  the  perioilicity  of  sin.T,  the  locus  of  ?/  =  sin  x 
consists  of  an  infinite  number  of  repetitions  of  the  curve 
obtained  from  a;  =  0  to  a;  =  2  tt.     The  locus  has  maximum  or- 

dinates    y  —  -{-1    corresponding   to   x  =  (4n  +  1)^, 


minimum 


ordinates  y  =  —  1  corresponding  to  x  =  (4  m  +  3)^,  where  n  is 

any  integer.     The  locus  crosses  the  a>axis  when  x  =  mr. 

To  plot  y—o  sin  a.",  it  is  only  necessary 
to  multiply  each  ordinate  oi  y  —  sin  x 
by  3.  This  is  effected  graphically  by 
drawing  a  pair  of  concentric  circles,  one 
with  radius  luiity,  the  other  with  radius 
3.  Since  OP'  is  the  linear  unit,  F'D'  rep- 
resents sin  X,  and  PD  represents  3  sin  x, 
*"^'  '^^'  while  X  is  represented  by  the  arc  A'P'. 

To  plot  v/=3  sin  a-+sin  (2  x),  plot  ?/i=3  sin  a;  and  ?/2=sin  (2  a-) 
on  the  same  axes.     The  ordinate  of  ?/  =  3  sin  x  +  sin  (2  .}•)  cor- 


N 

~ 

'h 

^ 

/ 

■\; 

; 

■\ 

\ 

/ 

% 

1/ 

v> 

\>, 

/ 

\ 

\ 

'''^ 

\ 

/  '• 

\ 

A 

I' 

\ 

\ 

v' 

^^ 

I'/ 

\ 

X 

\^ 

/j 

\^ 

\ 

/j 

' 

\ 

V 

ij 

\ 

i 

S 

/) 

\ 

y 

\ 

\, 

\ 

'/ 

V, 

' 

v 

responding  to  any  value  of  x  is  the  sum  of  t])o  ordinates  of 
yi  —  3  siiiiv  and  y.,  =  sin  (2.r)  corresponding  to  the  same  value 
of  X. 

When  the  sine-function  occurs  in  the  form  y  —  a  sin  (wt  +  6), 
where  w  is  uniform  angular  velocity  in  radians,  t  time  in  sec- 
onds, a  is  called  the  amplitude,  6  the  epoch  angle.    The  periodic 


PLOTTING    OF  THANSCEN DENTAL   EOUATIOXS       hi 


time  is  t  =  - —     Tlie  construction  of  the  curve  is  indicated  in 

the  iigure.     The  jtrojcction  of  unirorni  motion   in  the  circum- 
I'ci'ent'e  of  a  circle  on  a  diamctei'  is  caUcd  harmoiuc  motion. 


f=2 


t  =  ] 


1=0 


!J 

A 

r\ 

/ 

^-N 

\ 

/ 

\ 

/ 

\ 

\ 

/ 

\ 

\ 

/ 

\ 

t 

\ 

/ 

\ 

\ 

1 

\ 

\ 

1 

\ 

\ 

k/ 

1 

— 

— 

\ 

To  add  gi'apliically  two  sine-functions  of  equal  periods 
?/,  =  «i  sin  (co^  +  ^,),  ?/^  =  ao  (sin  w^  +  ^o),  draw  a  pair  of  con- 
centric circles  with  rndii  «,  and  a~y  Let  1\0D  and  P.,OD  be 
oj^  +  $1  and  (ot  -f  d.,  corresponding  to  the  same  value  of  (.    Then 


52 


ANAL  YTIC  GEOMETU F 


I\()D  -  P.,OD  =  ^1  -  e.,.  The  ijarallelogram  on  01\  and  01\ 
for  different  values  of  t  is  the  same  j)arallelogram  in  different 
positions.  This  parallelogram  has  the  same  angular  motion 
as  OPi  and  01\.     Now  y^  =  PiAj  2/2  =  P^D.,,  hence 

p/;-PiA  +  P.A  =  2/i  +  2/.. 

and  the  sum  of  the  sine-functions  corresponding  to  the  circular 
motions  of  Pi  and  P.,  is  the  sine-function  corresponding  to  the 
circular  motion  of  P.  The  resultant  sine-function  has  the  same 
period  as  the  component  sine-functions,  its  amplitude  is  OP, 
its  epoch  angle  the  angle  POD  corresponding  to  the  position 
of  P  for  t  =  0.  The  resultant  sine-function  is  y  =  a  s\n{wt-\-6), 
where  a^  =  a^^  +  a./  -\-2  ttia^  cos  {61  —  62), 

a  I  sin  $1  +  €(,2  sin  O2  * 
cij  cos  61  +  a.^  cos  62 

(1)  ij  =  sm~^x  is   equivalent  to  x 


tan  6 . 


is   equivalent  to   x  ■ 


sin  ^;    (o)  y  =  0  sin 
o 


sin  y ;    (2)  //  =  3  sin~'  x 
is  equivalent  to 


a;  =  3  sin:;;  (4)  /y  =  3  sii 


is  equivalent  to  x  —  2  sin 


graphic  interpretation  of  ccpuitions  (1),  (2),  (3),  (4)  is  shown 
in  figures  (a),  (b),  (c),  (d),  which  also  indicate  the  nu^nncr  of 
plotting  the  equations  graphically. 

*  A  jointed  paralk'lograin  is  used  for  conipoundiiig  harmonic  motions 
of  different  periods  in  Lord  Kelvin's  tidal  clock. 


PLOTTING   OF  TRANSCENDENTAL   EQUATIONS       53 

The  reiiuiining  circular  and  inverse  circular  fuuc^tions  are 
l)k)ttcd  in  a  manner  entirely  analogous  to  that  employed  in 
l)lotting  y  =  s'lnx  and  y  =  sin"'  x. 

Problems.  —  riot  1.  )j  =  2^.       2.  */ =  lO-'.      3.  y=(\y.      4.  y=(.l)^ 

5.  i/  =  2-^.    6.  .v  =  5-2^.    7.  y=3^\     8.  .v=:e^*    Q.  y  =  e-\    10.  ij  =  \(e^  +  e-^). 
This  function  is  called  the  hyperbolic  cosine,  and  is  written  y  =  cosh  a;. 

11.  y  —  -  (e^  +  e-^^)  or  y  =  c  cosh  x.    This  is  the  equation  of  the  catenary.t 

the  form  assumed  by  a  perfectly  flexible,  homogeneous  chain  whose  ends 
are  fastened  cat  two  points  not  in  the  same  vertical. 

12.  y  —  I  (e-^  —  e""^).     This  is  the  hyperbolic  sine,  and  is  written  sinh  r. 

13.  y  =  K 

14.  y  =  logiox. 

19.  ;K-2  =  logi„(y  +  5). 

20.  r  +  5  =  logio  (^  -  2). 


1.   y  =  logo  X. 

17.   2/  =  ;]log.,x. 

;.  ,  =  iog^,^_x. 

18.   2x  =  logio2/. 

26. 

»/  =  .3sinx. 

27. 

)/  =  sin  (4x). 

28. 

y  =  sm(lw  +  ^x). 

29. 

rj  —  3  +  sin  x. 

30. 

2/  =  3  +  sin(7r  +  2x). 

31. 

J/  =  sinx  +  2  sin  -■ 

22.   ^ ^  =  logio(y+l). 

X  +  a 

23    y  =  sin  ^^' 

■   ^  2  32.   2/  =  2  sin  (2  x)  +  3  sin  (3  x) . 

24.  ?/  =  .sin(2x).  33.   ?/  =  3sin(2  +  2x) +  fisin(l  +  4x). 

25.  ;/ =  sin  (x  +  .[ tt).  34.   y  =  cosx. 

35.  »/  =  5  cosx.  39.   ?/  =  secx.  43.  y  =  versx. 

36.  y  -2  cos  (1  +  Gx).    40.  y  =  2  secx.  44.  y  =  covers  x. 

37.  ?/  =  tanx.  41.   ?/ =  cosecx.  .     ,  ,, 

45.   X  =  sm-i  ^• 

38.  y  =  2  +  tan(l  +  x).   42.   y  =  cosec  (x  -  1).  2 

*  e  represents  the  base  of  the  Napierian  sy.stem  of  logarithms,  a  tran.scen- 
dental  number,  whose  value  to  nine  places  is  2.718281828.  y  —  e"  may 
be  plotted  graphically  by  computing  the  values  of  y  corresponding  to  any 
two  values  of  X  ;  numerically  by  writing  the  function  in  the  form  x  =  log,.»/ 
and  using  a  table  of  Napierian  logarithms. 

t  The  catenary  was  invented  by  Jolni  and  James  nernouUi.  The  center 
of  gravity  of  the  catenary  is  lower  than  for  any  other  position  of  the 
same  chain  with  the  same  fixed  points. 


54  ANALYTIC  GEOMETRY 

46.  2  M  =  cos-i  X.  -r  b2.  x  —  2  —  shr^  y. 

50.   w  =  3  cos-i  -• 

47.  2/ =  J  taii-i  x.  3  53.   x+3  =  siu-i  (i/-2). 

48.  2/  =  sec-i(x-3).       ^^    y  =  5sin-i?.  ^*-  ^  =  cos-i  (?/ -  1). 

49.  ?/ =  2  +  vers-i  X.  4  55.  2/+2=:cos-i(a;-2). 
56.  y  =  sill  (]  nt).                                  b1.   y  =  sin  (]  ivt  +  \  v). 

58.  y  —  sill  (.]  wt  +  2  t)  +  .siii(^  tt^  +  v). 

59.  ;/  =  sin  (I  nt)  +  sin  (}  irt  +  ^  tt). 

60.  y  =  sin  (|:  7r(  +  1  tt)  +  sin  {\  -ret  +  i  tt). 

The  elementary  transcendental  functions  are  of  great  impor- 
tance in  mathematical  physics.  For  instance,  if  a  steady 
electric  current  /flows  through  a  circuit,  the  strength  i  of  the 
current  t  seconds  after  the  removal  of  the  electromotive  force 

is  given  by  the  exponential  function  i=  le  ^,  where  R  and  L 
are  constants  of  the  circuit. 

The  quantity  of  light  that  penetrates  different  thicknesses  of 
glass  is  a  logarithmic  function  of  the  thickness. 

The  sine-function  is  the  element  by  whose  composition  any 
single-valued  periodic  function  may  be  formed.  Vibratory 
motion  and  wave  motion  are  periodic.  The  sine-function  or, 
as  it  is  also  called,  the  simple  harmonic  function,  thus  becomes 
of  fundamental  importance  in  the  mathematical  treatment  of 
heat,  light,  sound,  and  electricity. 


Art.  26.  —  Cycloids 

A  circle  rolls  along  a  fixed  straight  line.  The  curve  traced 
by  a  point  fixed  in  the  circumference  of  the  circle  is  called  a 
cycloid.*  The  fixed  line  is  called  the  base,  the  point  whosr 
distance  from  the  base  is  the  diameter  of  the  generating  circli^ 
the  vertex,  the  perpendicular  from  the  vertex  to  the  base  th<' 

*  Curves  generated  by  a  point  fixed  in  the  plane  of  a  curve  which  \\A[a 
along  some  fixed  curve  are  called  by  the  general  name  "roulettes.'' 
Cycloids  are  a  special  class  of  roulettes. 


a  XI 
in 


P LOTTING    OF  TRANSCENDENTAL    EQCATlONH       .55 

mS  of  tlicryi'loiil.     The  courtliiiates  of  any  inuiitof  the  cycloid 
ay  be  expressed  as   trausceudental   i'uiictions  cd'  a   variable 
angle. 

Take  the  base  ol'  the  cyck)id  as  X-axis,  the  perpendicular  to 
the  base  where  the  cycloid  meets  the  base  as  I'-axis,  and  call  the 
angle  made  by  the  radius  of  the  generating  circle  to  the  tracing 
point  with   the  vertical   diameter  6.     l>y  the   nature  of   the 


cycloid  AK=  arc  PK  =  rO,  y  =  PD  =  OK  -  OL  ^  r  -  r  cos  d, 
X  =  AD  =  AK  —  DK=^  i-e  —  r  sin  9.  Kence  (1)  x  =  rO  -  r  sin  0, 
y  =  )•  —  r  cos  9  for  every  value  of  9  determine  a  point  of  the 
cycloid.  The  equation  of  the  cycloid  between  x  and  y  is 
obtained  either  directly  from  the  figure, 

X  =  AK  —  DK  —  arc  PK  —  PL  =  r  vers"'* —  Vli  ni 

r 

or  by  eliminating  9  between  ecpuitions  (1), 


T 


V 


.  V -5  ry  -  y-, 


hence 


Vi/ 


ry 


Now  r  vers~^-^  has  for  the  same  value  of  y  an  infinite  number 


of  values  differing  by  2tt,  and  V2ry  —  //  is  a  two-valued  func- 
tion which  is  real  only  for  values  of  y  from  0  to  -f  2  r.  Hence 
the  equation  determines  an  infinite  number  of  values  of  x  for 
every  value  of  y  between  0  and  +  2  r.  This  agrees  with  the 
nature  of  the  curve  as  determined  by  its  generation. 


56 


A  NA L  YTIC  GEOMETR  Y 


By  observing  that  tlie  center  of  the  generating  circle  is 
always  in  the  line  parallel  to  the  base  at  a  distance  equal  to 
the  radius  of  the  generating  circle,  the  generating  circle  may 
readily  be  placed  in  position  for  locating  any  point  of  the 
cycloid.  At  the  instant  the  point  P  is  being  located  the 
generating  circle  is  revolving  about  K,  hence  the  generating 
point  P  tends  at  that  instant  to  move  in  the  circumference  of 
a  circle  whose  center  is  A' and  radius  the  chord  KP.  The  tan- 
gent to  the  cycloid  at  P  is  therefore  the  perpendicular  to  the 
chord  KP  at  P,  that  is  the  tangent  is  the  chord  PII  of  the 
generating  circle.* 

The  perpendicular  to  the  tangent  to  a  curve  at  the  point  of 
tangency  is  called  the  normal  to  the 
curve  at  that  point.  Hence  the  chord 
KP  is  the  normal  to  the  cycloid  at  P. 

Take  the  axis  of  the  cycloid  as  X-axis, 
the  tangent  at  the  vertex  as  F-axis. 
By  the  nature  of  the  cycloid 

3//ir=arcP/i, 

MN—  semi-circumference  KPH. 

X  =AD  =  HL=OH-OL  =  r-r cos  0, 

7j  =  PD  =  LD+PL=(MN-Miq-\-PL 

=  arc  IIP  +  PL=  rO  +  r  sin  0. 


That  is, 


(1)   X: 

y 


rd  -\-  r  sin  i 


determine  for  every  value  of  ^  a  point  of  the  cycloid.  The 
equation  between  x  and  y  is  found  either  directly  from  the 
figure, 

y  =z  LD  +  PL  =  r  vers"'  -  +  V2  rx  —  x- ; 


This  method  of  drawing  a  tangent  to  the  cycloid  is  due  to  Descartes. 


V LOTTING   OF  THAN SiCEN DENTAL   EQUATIONS       57 

or  by  elimiiiatini;-  6  between  eiiuations  (1), 

^=:cos-/l  --")=:  vers-' ^, 


.6^^V2' 


hence 


y=r  vers"'  -  +  V2  rx 
r 


Art.  27.  —  Prolate  and  Curtate  Cycloids 

When  the  genercating  point  instead  of  being  on  the  eircuni- 
ference  is  a  point  fixed  in  the  pkme  of  the  rolling  circle,  the 
curve  generated  is  called  the  prolate  cycloid  when  the  point  is 
within  the  circumference,  the  curtate  cycloid  when  the  point 
is  without  the  circumference.  Let  a  be  the  distance  from  the 
center  to  the  generating  point.  From  the  figures  the  equations 
of  these  curves  are  readily  seen  to  be 

x  =  r6  —  a  sin  0,   y  =  r  —  a  cos  6. 


Fio.  40 


*  If  the  cycloid  is  concave  up  and  the  tanfjcnt  at.  the  vertex  horizontal, 
the  time  required  by  a  particle  sliding  down  the  cycloid,  suiJjiosed  friction- 
less,  to  reach  the  vertex  is  independent  of  the  starting  point.  On  account 
of  this  property,  discovered  by  Huygens  in  1G73,  the  cycloid  is  called  the 
tautochrone.  The  frictionless  curve  along  which  a  body  must  slide  to 
pass  from  one  point  to  another  in  the  shortest  time  is  a  cycloid.  On 
account  of  this  property,  discovered  by  John  Bernoulli  in  1G9G,  the  cycloid 
is  called  the  brachistochrone. 


58 


ANALYTIC  GEOMETRY 


Art. 


Epicycloids  and  Hypocycloids 


If  a  circle  rolls  along  the  circumference  of  a  fixed  circle,  the 
curve  generated  by  a  point  fixed  in 
the  circumference  of  the  rolling 
circle  is  called  an  epicycloid  if  the 
circle  rolls  along  the  outside,  an 
hypocycloid  if  the  circle  rolls  along 
the  inside  of  the  circumference  of 
the  fixed  circle.  By  the  nature  of 
Oi  ~U  D  X  the  epicycloid  arc  HO  =  arc  HP, 
that  is  E-6^r-  (f>.  From  the 
figure  X  =  AD  =  AL  +  DL 
Fiu.  47.  =(R  +  r)cos  0  +  r  cos  CPM. 

...  Ji  +  r.      „  _ 


180°-: 


Hence  x  =  (R  +  r)cos  0 


PD  =  CL 

R  +  r 


CM 


=  (R  +  r)sin^  —  r  sin 

By  the  nature  of  the  hypocycloid  R  •  6 

=  r  •  <^,    hence  ^  =  ^-6.     x  =  AD 
r 

X    =  AL  -  PM 

=  (R  -  r)cos  e-  7-  cos  CPM. 

NowCPJ/=lSO°-</,  +  ^ 
R- 


x=(R  —  r)cos  6  +  r  cos 


180° 
R- 


Hence 


PD  =  CL  -  CM  =  (R-  r)s\n  O-r  sin 


Epicycloids  and  hypocycloids  are  used  in  constructing  gear  teeth. 


PLOTTING   OF  TRANSCENDENTAL   EQUATIONS       f)!) 

AiiT.  29.  —  Involutk  of  Cikcli': 

A  string  whose  length  is  the  circumference  of  a  circle  is 
wound  about  the  circumference.  One  end  is  fastened  at  0  and 
the  string  unwound.     If  the  string  is  kept  stretched,  its  free 


end  traces  a  curve  which  is  called  the  involute  of  the  circle. 
From  the  nature  of  the  involute,  IIP  is  tangent  to  the  fixed 
circle  and  equals  the  arc  HO,  which  equals  Ji$. 

X  =  AD  =  AL  +  KP  =  11  cos  0  +  BO  sin  6, 

y=PD=  IIL  -  IIK  =  n  sin  6  -  116  cos  6* 

*  The  invohite  is  also  used  in  cnnstructing  gear  teeth. 


CHAPTER   V 


TEANSrOEMATION  Or  COOKDINATES 

Art.  30.  —  Transformation  to  Parallel  Axes 

Let  P  be  any  point  in  the  plane.  Keferred  to  the  axes 
X,  Y  the  point  P  is  represented  by  {x,  y) ;  referred  to  the 
parallel  axes  Xj,  Yi,  the  point 
P  is  represented  by  (x^,  y-^.  Let 
the  origin  A^  be  {m,  n)  when 
referred  to  the  axes  X,  Y. 
— Xi  Prom  the  figure  x  =  m  +  x^, 
y  =  n  -{-  yi.  Since  (x,  y)  and 
— X  (xi,  ;Vi)  represent  the  same  point 
P,  if  f{x,  ?/)  =  0  is  the  equation 
of  a  certain  geometric  figure 
when  interpreted  with  refer- 
ence to  the  axes  X,  Y,  f(m  +  Xi,  n  -|-  ?/i)  =  0  is  the  equation  of 
the  same  geometric  figure  when  interpreted  with  reference  to 
the  axes  Xj,  Y^. 

Example.  —  The  eqiiation  of  the 
circle  whose  radius  is  5,  center  (2,  3) 
is  (1)  (a;  -  2)2  +  (y  -  3)^  =  25.  Draw 
a  set  of  axes  Xj,  Yi  parallel  to  X, 
Y  through  (2,  3).  Then  x  =  2  +  x„ 
y  =  ?y  -\-  ?/,.  Substituting  in  equation 
(1),  there  results  (2)  x^-  +  y,^  =  25. 
Notice  that  the  equation  of  a  geo- 
metric figure  depends  on  the  position 
of  the  geometric  figure  with  respect 
to  the  axes. 
GO 


TltANSFOIiMATION    OF  COOHDINATES 


Gl 


AUT.    31.  —  FUOM    KeCTANGULAU    AxKS    to    JiK(  TAN(i[M.AIl 

Let  (x,  y)  represent  any  point  in  the  plane  referred  to  the 
axes  X,  F;  (x„  y^  the  same  point  referred  to  the  axes  X,,  \\, 
where  Xj,  Yi  are  obtained  by  turning  A',  Y  about  A  througli 
the  angle  a.     Now 

x=AD  =  An-KD' 

=  x'l  cos  a  —  .'/i  sin  a, 
y=.  P1)  =  D'H+PK 

—  .Xisin  a  +  yi  cos  a. 

Since  (.x*,  .7)  and  (.i-,,  yO  rep- 
resent the  same  point  7*, 
/(x,  ?/)  =  0  interpreted  on  the 
A",  Y  axes  and  r><i.  52. 

/(a'l  cos  a  —  ?/i  sin  «,    a;,  sin  «  +  ?/,  cos  «)  =  0 
interpreted  on  the  Xj,  Fj  axes  represent  the  same  geometric 
figure. 

Example.  — y  =  a-  +  4  is  the  equation  of  a  straight  line.  To 
find  a  set  of  rectangular  axes,  the  origin  remaining  the  same,  to 
which  when  this  line  is  referred 
its  equation  takes  the  form  yi=n, 
substitute  in  the  given  equation 

X  =  Xi  cos  a  —  yi  sin  a, 
y  =  Xi  sin  a  +  y^  cos  a. 
There  results 

Xy  (sin  a  —  cos«) 

+  ?/i  (sin  a  +  cos  «)  =  4, 

and  this  equation  takes  the  re- 
quired form  when 

sin  a  —  cos  a  =  0, 

that  is,  when  a  =  45°.     Substituting  this  value  of  a,  the  trans- 
formed equation  becomes  ?/,  =  2  V2. 


ANALYTIC   GEOMETRY 


Art.  32.  —  Oblique  Axes 
Hitherto  the  axes  of  reference  have  been  perpendicular  to 
each  other.    The  position  of  a  point  in  the  plane  can  be  equally 
well  determined  when  the  axes  are  oblique.    The  ordinate  of  the 

point  F  referred  to  the 
/  /  /  oblique  axes  X,  Yis  the 
distance  and  direction 
of  the  point  from  the 
X-axis,  the  distance 
being  measured  on  a 
parallel  to  the  I''-axis, 
the  side  of  the  X-axis 
on  which  the  point  lies 
being  indicated  b}^  the 
algebraic  sign  prefixed 
" '"' ""'  to  the  number  express- 

ing this  distance.  Similarly  the  abscissa  of  the  point  P  is  the 
distance  and  direction  of  the  point  from  the  F-axis,  the  dis- 
tance being  measured  on  a  parallel  to  the  X-axis,  the  alge- 
braic sign  prefixed  to  this  distance  denoting  on  what  side  of 
the  F-axis  the  point  lies. 

Problems.  — Tlie  angle  between  the  oblique  axes  being  45' : 

1.    Locate  the  points  (3,  -  2);  (-  5,  4);  (0,  8);  (-  4,  -  7);  (2i,  -  3); 


Observe  that  the  geometric 
figure  represented  by  an  equa- 
tion depends  on  the  system  of 
coordinates  used  in  plotting  the 
equation. 

4.  Find  the  equation  of  a 
straight  line  referred  to  oblique 
axes  including  an  angle  /3.  The 
method  used  to  find  the  equation 
of   a   straiafht  lino  referred   to 


( 

v/5, 
2. 

-V7);   (-21,  V 
Plot  2/  =  3  X  ;  ?/  = 

'10) 
3x 

+  5; 

y  = 

-2x 

3. 

Plot  x^  +  y^^  IG 

;  !/■ 

=  4x 

.  x^ 
'   9 

-!  = 

TRAysFOnMAriON   OF  COOIiDrXATES 


03 


rcctancjular  axes  sliows  tliat  the  equation  of  a  strai,i,'lit  line  referred  to 
oblique  axes  is  y  =  vix  +  «,  where  in  — 


of  the  line  on  the  I'-axis. 


sni  (/S  —  a) 


and  n  is  the  intercept 


5.  Show  that  V  (x'  -  x")'^  +  (y'  -  y")'^  +  2(3;'  -  x")  {y'  -  y")  cos  /3  is 
the  distance  between  the  points  (z',  ?/'),  (x",  y")  when  the  angle  between 
the  axes  is  /3. 

6.  Find  the  equation  of  the  circle  whose  radius  is  Ji,  center  (m,  7i), 
when  the  angle  between  the  axes  is  /3. 

7.  Show  that  double  the  area  of  the  triangle  whose  vertices  are 

(3^1,  yO,  (3-2,  !/2),  (a^s,  2/3) 
is  {yi(^3  -  X2)  +  yo(xi  -  T:i)  +  y3(x-2  -  xi)}sin  /3. 

AuT.  33.  —  From  Rectangular  Axes  to  Oi-.lique 

It  is  sometimes  desirable  to  find  tlie  equation  of  a  £;eometric 
figure  referred  to  oblique  axes  when  the  equation  of  this  figure 
referred  to  rectangular  axes  is 
known.  This  manner  of  ob- 
taining the  equation  of  a  figure 
referred  to  oblique  axes  is  fre- 
quently a  simpler  problem  than 
to  obtain  the  equation  directly. 
To  accomplish  the  transformation, 
the  rectangular  coordinates  of  a 
point  must  be  expressed  in  terms 
of  the  oblifjue  coordinates  of  the  Fio.  5g. 

same  point.     From  the  figure 

a:  =  AD  =  AII+  D'K=  .r,  cos  a  +  ,v,  cos  a', 
y  =  PD=  D'II+  PK=  .r,  sin  a  +  //,  sin  u'. 

ExAMi'LE.  —  To  find  the  equation  of  the  h_viHTl)i)l;i  rcfcvvcd 
to  its  asymptotes  from  the  common  C(piatit)n  of  the  hyperbola, 
x^  _  jf  _  1 


64 


ANALYTIC  GEOMETin 

b 


The  asymptotes  of  the  hyperbola,  y 
of  the  rectangle  on  the  axes.     Hence 
b 


±  -X,  are  the  diagonals 
a 


cos  a  —      

Va-  +  b- 

and  the  transformation  for- 
mulas become 

"      (.-^-i  +  z/O, 


Va^  +  b' 
b 


Va^  +  6- 
Substituting  in  the  common 
equation   of    the    hyperbola 

and   reducing,  a\y^  = '-, 

4 
the  equation  of  the  hyperbola  referred  to  its  asymptotes. 

The  formulas  for  passing  from  oblique  axes  to  rectangular, 
'y  the  origin  remaining  the  same,  are 

x  =  AD  =  AH-D'K 

_  y,  sin  (/3  —  a)  _  ?/,  cos  ((3  —  a) 
sin/?  sin/3 

=  PD  =  D'll  +  PK 

_  Xi  sin  a     ?/i  cos  n 
sin  /3         sin  (3 


Art.  34.  —  General  Transformation 

The  general  formulas  for  transforming  from  one  set  of  recti- 
linear axes  to  another  set  of  rectilinear  axes,  the  origin  of  the 
second  set  when  referred  to  the  first  set  being  (m,  n),  are 


TUANSF(>i;MAril)i\    OF   I'OOUDl X ATKS 


05 


x  =  AD=An  +  A/r-\-J)'h', 

_  .r I  s i  11  (/3  -  u)      //i  sill {13 -a') 

-'"+         sm(3         ^         sm(3        ' 
2/  =  PL*  =  .l,7t*  +  D'T  +  I'K, 

.r,  sill  u  ,   Vi  sill  a' 


From  tliesc  L;eiieral  formulas  all  tlie  ijrecediiiy  formulas  may 
bo  derived  by  substituting  for  ?/;,  v,  fS,  a,  a'  tlieir  values  in 
each  special  case.  However,  if  it  is  observed  that  in  every 
case  the  figure  used  in  deriving  the  transformation  formulas 
is  constructed  by  drawing  the  coordinates  of  any  ])oint  P 
referred  to  the  original  axes,  and  the  coordinates  of  the  same 
point  referred  to  the  new  axes,  then  through  the  foot  of  the 
new  ordinate  parallels  to  the  original  axes,  it  is  simpler  to 
derive  these  formulas  directly  from  the  figure,  whenever  they 
are  needed. 


Art.  35.  —  Thk  ruouLEM  of  Tiia\sfoi;i\i.\tion 

An  examination  of  the  transformation  formulas  shows  that 
the  values  of  the  rectilinear  coordinates  of  any  point  in  terms 
of  any  other  rectilinear  coordinates  of  the  same  ])oint  are  of 
the  first  degree.     Hence  transformation  from  one  set  of  recti- 


66  ANALYIUC  GEOMETRY 

linear  coordinates  to  auotlier  rectilinear  set  does  not  change 
the  degree  of  the  equation  of  the  geometric  figure. 

Two  classes  of  problems  are  solved  by  the  transformation  of 
coordinates : 

I.  Having  given  the  equation  of  a  geometric  figure  referred 
to  one  set  of  axes,  to  find  the  equation  of  the  same  geometric 
figure  referred  to  another  set  of  axes. 

II.  Having  given  the  equation  of  a  geometric  figure  referred 
to  one  set  of  axes,  to  find  a  second  set  of  axes  to  which  when 
the  geometric  figure  is  referred  its  equation  takes  a  required 
form. 

Problems.  —Transform  to  parallel  axes,  given  the  coordinates  of  the 
new  origin  referred  to  the  original  axes. 

1.  2/  -  2  =  0(x  +  5),  origin  (-  5,  2). 

2.  (x  -  3)  (?/  -  4)  =  5,  origin  (3,  4). 

3.  y  =  2x  +  5,  origin  (0,  5). 

4.  a;2  +  ?/2  4-  2  X  +  4  ?/  =  4,  origin  (  -  1,  -  2). 

5.  a;2  +  2/2  +  fi  »/  =  7,  origin  (0,  -  3). 

6.  X-  +  2/2  -  G  X  =  16,  origin  (3,  0). 

7.  2/^  +  4  2/  -  G  X  =  4,  origin  (0,  -  2). 

8.  25(2/  +  4)2  +  lC(x  -  5)2  =  400,  origin  (5,  -  4). 

9.    ,;2  +  f.  ^  25,  origin  ( -  5,  0).         ^^     ^J  +  ?^  =  1,  origin  (0,  -  h). 

10.  x2  +  2/2  =  25,  origin  (0,  -  5).  "'      ''' 

11.  x2  + 2/2  =  25,  origin  (-5,  -5).    '^-    |  +  g  =  1,  origin  (- a,  -  6). 

12.  ^^t^  1,  origin  (-  a,  0).  15.    ^-^-  =  1,  origin  (a,  0). 
a-     62  a^     h^ 

Transform  from  one  rectangular  set  to  a  second  rectangular  set,  the 
second  set  being  obtained  by  turning  the  first  about  the  origin  tlirough  45'^. 

16.  x2  +  2/-  =  4.  18.    y  +  x  =  5.  21.    y'^-3ry  +  x«,=  0. 

17.  x2  -  2/2  =  4.  19.    2/2  ^  jy  _  a:2  =  G.  22.    ?/"  +  3  ry  -  .r'  =  0. 

20.    if'  +  4  xy  +  x2  =  8. 

Notice  that  to  plot  equations  21  and  22  directly  requires  tlie  sohition  of 
a  cubic  equation,  whereas  the  transformed  equations  are  plotted  by  the 
solution  of  a  quadratic  equation. 


TliANSFORMATION   OF  COORDINATES 


(37 


In  the  following  problems  the  first  equation  is  the  equation  of  a  geo- 
metric figure  referred  to  rectangular  axes.  The  origin  of  a  parallel  set  of 
axes  is  to  be  ftiuml  to  which  when  the  geometric  figure  is  ri'ferred  its 
equation  is  tlie  second  eiiuation  given. 

23.  .'/  +  2  =  4(x  -  :J) ;  y  =  -1  •^■-  26.    if  -  X-  -  lU  x  =  0  ;  x-  -  ij-  =  25. 

24.  (.<•  +  1)  (//  +  5)  =  4  ;  xij  ^  1.        27.    U'  -  1U(..:  -|-  5)  =  0  ;  >f  =  10  x. 

25.  f  +  x^  +  lOx=0;  x:-  +  y'=25.      28.    if  +  x;^  +  -i  >j--2  x=l\  ;  x-  +  f  =  YG. 

In  the  following  problems  the  first  equation  is  tlu;  equation  of  a  geo- 
metric figure  referred  to  rectangular  axes;  find  the  inclination  of  a  secoiul 
set  of  rectangular  axes  to  the  given,  origin  remaining  the  same,  to  which 
when  the  geometric  figaire  is  referred  its  equation  is  the  second  eiiuation 
given. 
29.    y  =  x  +  4;  y  =  2V2.  ^^ 

„o  ^  2n  ■    ,-2  4-  y2  =  2:,.  32.    ^  -  2g  =  1  ;  :r// 


30. 


y- 


33.    //■'  -  3  axy  +  .';'  =  0  ;   y^ 


-  y 
3V2aa;' 


d'  4-  }i~ 


2x3 


2x  -|-;j\/2a 

Construct  the  locus  of  tlie  first  etiuation  in  the  following  problems  by 
drawing  tlie  axes  A'l,  I'l  and  plotting  the  second  eipiation. 

34.  11-2  =  log(x  -f-  ;:i) ;  ?/i  =  log  Xi.    36.    ?/  -f  3  ==  2^  t  •» ;  y.^  =  2a. 

35.  2/  =  3  sin(x  +  5);  2/1  =  3  sin  Xi.     37.    y  +  5  =  tau(x  -  3);  yi  =  tan  Xi. 

1,  obtain 


3.2      ,,2 
From  the  common  eciuation  of  the  hyperbola,-—  —  -- 


the  equation  of  the  hyperbola  referred  to  oblique  axes  through  the  center, 

1,2 

such  that  tan  a  tan  a'  =  — 

a- 

The   transformation    formulas 

are    x  =  Xi  cosa -f- 2/1  cosa', 

y  =  Xisina  -1-  yi  sin  a' ; 

the  transformed  equation 

/cos^a      sin-a\     , 

[—2  ^p' 

+  2f"°^°' 


sm  g  sm  g' 

The  condition  tan  a  tan  a 
the  equation  of  the  hyperbola  referred  to  the  oblique  axes  becomes 


Fic.  f>i». 
renders  the  coefficient  of  Xij/i  zero,  and 


68 


ANALYTIC  GEOMETRY 


.siii-c 


l.'/r  =  1- 


Since  only  values  uf  a  and  a'  less  than  180°  need  be  considered,  the  con- 
dition tan  a  tan  a'  =    -  shows  that  a  and  a'  are  either  both  less  than  90°  or 

«"  h  b 

both  greater  than  90",  and  that  if  tana  <-,  tana' >-.  Since  the  equa- 
tions of  the  asymptotes  of  the  hyperbola  are  y  =  ±~x,  it  follows  that  if 
the  Xi-axis  intersects  the  hyperbola,  the  Ti-axis  cannot  intersect  it. 
Calling  the  intercepts  of  the  hyperbola  on  the  Xi-  and  I'l-axis  respectively 
ai  and  biV—  1,  the  equation  referred  to  the  oblique  axes  becomes 


39.   From  the  common  equation  of  the  ellipse,  ~  -|-  -  =  1,  obtain  the 


62 


62 


P  P 

(m,  n)  is  y=~(x  +  7n),  tano'=— . 

n  n 


equation  of  the  ellipse  referred  to  oblique  axes  such  that  tan  a  tan  a' 

40.  From  the  common  equation  of  the  parabola  y^  =  2px  obtain  the 
equation  of  the  parabola  referred  to  oblique  axes,  origin  (m,  n)  on  the 
parabola,  the  AVaxis  parallel  to  the  axis  of  the  parabola,  the  JVaxis 
tangent  to  the  parabola. 

n-  =  2  pm,  a  =  0,  and,  since  the  equation  of  the  tangent  to  y"^  =  2px  at 
The  transformation  formulas  be- 
come X  =  m  +  Xi  +  ?/i  cos  a',  ?/  =  n  +  ?/i  sin  a',  and  the  transformed  equa- 

rfl  -|-   /i'.2 

tion  reduces  to  yi^  =  2 .ti,  or  yi~  ■=  2(p  +  2  m)xi. 

41.  To  determine  a  set  of  oblique  axes,  with  the  origin  at  the  center,  to 
which  when  the  ellipse  is  referred,  its  equation  has  the  same  form  as  the 
common  equation  of  the  ellipse  -^  +  ra  =  1- 

The  substitution  of 

X  =  a:i  cosa  +  ?/iC0Sa', 
2/  =  Xi  sin  a  -1-  yi  sin  a' 

transforms  the  equation  -2+72  =  1 

into 

/Cos2a 


V    rt2 


+ 


ft2 


TRANSFOh'MATIO.Y   OF  COORDINATES  G9 

'I'lio  problem  requires  thai  the  coefficient  of  Xij/i  bo  zero,  hence 

tan  a  tan  a.'  = '- 

d- 

Tlie  problem  is  indeterminate,  since  the  etiualion  between  a  and  a'  admits 
an  infinite  number  of  solutions.  Let  a  and  a'  in  the  figure  represent  one 
solution,  then  (^ +  ^^^^^-3  +  ^^' +  ^li^^y,.  ^  1  is  the  equa- 
tion  of  the  ellipse  referred  to  the  axes  A'l,  I'l.  Call  the  intercepts  of  the 
ellipse  on  the  axes  Xi  and  I'l  respectively  ai  and  hi ,  and  the  equation 
becomes  ^  +  f^  =  1. 

When  the  equation  of  the  ellipse  referred  to  a  pair  of  lines  through  the 
center  contains  only  the  squares  of  the  unknown  quantities,  these  lines 
are  called  conjugate  diameters  of  the  ellipse.     The  condition  of  conjugate 

diameters  of  the  ellipse  is  tan  a  tan  a'  =  —  — . 

a' 

42.  Determine  a  set  of  oblique  axes,  with  the  origin  at  the  center,  to 
which,  when  the  hyperbola  is  referred,  its  equation  takes  the  same  form 
as  the  common  equation  of  the  hyperbola. 

The  result,  tanatana'  =  — ,  shows  that  the  problem  is  indeterminate. 
jfl  «'- 

tana  tan  a'  =  —  is  the  condition  of  conjugate  diameters  of  the  hyperbola. 
d^ 

43.  Determine  a  set  of  oblique  axes,  origin  at  center,  to  which,  when 
the  hyperbola  is  referred,  its  equation  takes  the  form  xij  —  c. 

44.  Determine  origin  and  direction  of  a  set  of  oblique  axes  to  which, 
when  the  parabola  is  referred,  its  equation  has  the  same  form  as  the 
common  equation  of  the  parabola. 

45.  Show  that  the  equation  of  the  parabola  y'^  =  2pz  when  referred  to 
its  focal  tangents  becomes  x^  +  y^  —  a^,  where  a  is  the  distance  from  the 
new  origin  to  the  points  of  tangency. 


CHAPTER   VI 


POLAR   OOOEDINATES 


Art.  36. — Tolau  Cooudinates  of  a  Point 

In  the  plane,  suppose  the  point  A  and  the  straight  line  AX 
through  A  fixed.     A  is  called   the  pole,  AX  the  polar  axis, 
p ,/  The  angle  which  a  line  AP  makes 

^'"''^^  with  AX  is  denoted  by  6.  0  is 
positive  when  the  angle  is  con- 
ceived to  be  generated  by  a  line 
X  starting  from  coincidence  with  AX 
turning  about  A  anti-clockwise ;  9 
is  negative  when  generated  by  a 
line  turning  about  A  clockwise. 
When  6  is  given,  a  line  through  A 
is  determined.  On  this  line  a  point  is  determined  by  giving 
the  di-stance  and  direction  of  the  point  from  A.  .  The  direction 
from  A  is  indicated  by  calling  distances  measured  from  A  in 
the  direction  AP  of  the  side  of  the  angle  0  positive,  those 
measured  in  the  opposite  direction  negative.  The  point  P  is 
denoted  by  the  symbol  (r,  ^),  the  point  P'  by  the  symbol 
(-r,  e).  The  symbols  (r,  ^  +  27r?i),  (-'',  ^ +(2?i  +  l)7r), 
where  n  is  any  integer,  denote  the  same  point.  To  every  sym- 
bol (r,  &)  there  corresponds  one  point  of  the  plane ;  to  every 
point  of  the  plane  there  corresponds  an  infinite  number  of 
symbols  (r,  d).  Under  the  restriction  that  r  and  6  are  positive, 
and  that  the  values  of  B  can  differ  only  by  less  than  2  tt,  there 
exists  a  one-to-one  correspondence  between  the  symbol  (/•,  Q) 
70 


POLAR    coon  DIN  ATES  71 

aud  the  points  of  the  plane,  the  pole  only  excepted,     r  and  0 
are  called  the  polar  coordinates  of  the  point. 

1.  Locate  the  points  whose  polar  coordinates  are  (2,  0) ; 
(-3,0);  (3,l,r);  (-2,7r);  (4,f,r);  (-4,l7r);  (4,-|7r);  (1,1); 
(-2,1);  (-1,0);  (1,180°);  (-4,45°);  (4,225°);  (-4,405°); 
(0,  0) ;  (0,  45°) ;  (0,  225°). 

2.  Show  that  r'-  +  r"-  -  2  r'r"  cos  (0'  -  0")  is  the  distance  be- 
tween (?•',  6'),  (r",  0"). 

3.  Find  the  distances  between  the  following  pairs  of  points, 
(4,l7r),  (3,7r);  (8,J-7r),  (6,f,r);  (2V2,  -^7^),  (l,i7r);(0,0), 
(10,  45°);  (5,  45°),  (10,  90°) ;  (-0,  120°),  (-  8,  30°). 

AuT.  37. — PoLAK  Equations  ok  CiEoiiETiiio  Fiuujies 

The  conditions  to  be  satisfied  by  a  moving  point  can  some- 
times be  more  readily  expressed  in  polar  coordinates  than  in 
rectilinear  coordinates.  If  a  point  moves  in  the  XF-plane  in 
such  a  manner  that  its  distance  from  the  origin  varies  directly 
as  the  angle  included  by  the  X-axis  and  the  line  from  the 
origin  to  the  moving  point,  the  rectangular   equation  of   the 

locus  is  Va.'-  -f  y-  —  a  tan~^-,,  the  polar  equation  r  =  a9. 

Desired  information  about  a  curve  is  often  obtained  more 
directly  from  the  polar  equation  than  from  the  rectilinear 
equation  of  the  curve.  This  is  especially  the  case  when  the 
distances  from  a  fixed  point  to  various  points  of  the  curve  are 
required.  Thus  if  the  orbit  of  a  comet  is  a  parabola  with  the 
sun  at  the  focus,  the  comet's  distance  from  the  sun  at  any  time 
is  obtained  directly  from  the  polar  equation  of  the  parabola. 

AuT.  38.  —  Polar  Equation  of  Stuakjut  Link 

A  straight  line  is  determined  when  tlie  lengtli  of  tlic  perpen- 
dicular from  the  pole  to  the  line  and  the  angle  included  by  this 
perpendicular   and   the  polar  axis  are  given.     Call  the  per- 


ANAL  YTIC   GEOMETR Y 


peiuliciilar  p,  the  angle  a,  and  let  (r,  6)  be  any  point  of  the 
line.     The  equation 


Fig.  C3.  \ 

;    for    6  =  a. 


cos  {p  —  a) 
expresses  a  relation  satisfied  by 
the  coordinates  r,  0  of  every  point 
of  the  straight  line  and  by  the 
coordinates  of  no  other  point; 
that  is,  this  is  the  equation  of 
the    straight     line.       For    6  —  0, 


from   ^  =  0   to   ^  =  90°  +  «,    r   is 
cos  «  ' 

positive  ;  from  0  =  00°  +  a  to  0  =  270°  +  a,  r  is  negative  ; 
from  6  =  270°  +  a  to  6  =  3G0°,  r  is  again  positive.  For 
6  =  90°  +  a  and  for  6  =  270°  +  «,  r  =  ±  oo  .  These  results  ob- 
tained from  the  equation  agree  with  facts  observed  from  the 
figure. 

A  straight  line  is  also  determined  by  its  intercept  on  the 
polar  axis  and  the  angle  the  line  makes  witii  the  polar  axis. 

Call    the    intercept    b,    the 
v'^-'  angle  a,  and  let  (r,  6)  be  any 

point    in    the    line.      Then 

6  sin  a       .     ^, 

r  =—. — 7 ;rr    IS   thc    cqua- 

sin(«  — ^) 

tion  of  the  line.     For  ^  =  0, 

r  =  b;     ?•    is    positive    from 

^  =  0    to    6  =  a;     negative 

from  ^  =  a  to  6  =  180°  +  a; 

^"'-  ^-  again  positive  from  6  =  180° 

+  atoe  =  360°.     ¥ov  0=a  and  6  =  180°  +  «,  r  =  ±  oo.     These 

results  may  be  obtained  from  the  equation  or  from  the  figure. 

Akt.  39.  —  Polar  Equation  of  Circle 

The  equation  of  a  circle  whose  radius  is  E  when  the  pole  is 
at  the  center,  the  polar  axis  a  diameter,  is  r  =  E. 


POLAR   COOUDl NATES 


73 


AVheu  tlic  pole  is  on  tlio  circuinfeieuce,  tlio  polar  axis  a 
diameter,  the  equation  of  the  circle  is  r  =  2  li  cos  0. 

r  is  positive  from  ^  =  0°  to  ^  =  90°,  negative  from  $  =  90°  to 
6  =  270°,  and  again  positive  from  6  =  270°  to  ^  =  300°.  The 
entire  circumference  is  traced  from  ^  =  0°  to  6  =  180°,  and 
traced  a  second  time  from  ^  =  180°  to  ^  =  ,'>60°. 


The  polar  equation  of  a  circle  radius  R,  center  (>•',  6'),  cur- 
rent coordinates  r,  6,  is  »-^  —  2  r'r  cos  (6  —  6')=  lir  —  r'-,  whence 
r  =  r'  cos  (6  -  $')  ±  -y/R-  -  r'-  sm\e  -  J'),  r  is  real  and  has 
two   unequal    values   when    sin-(^  —  ^')< -y,;    that    is,   wIkmi 

R       ■  R  ^ 

<sin(^  — ^')< — ;    these  values  of  r  become  equal,  and 

r  r  jy 

the  radius  vector  tangent  to  the  circle,  when  sin(^  — ^')=  ±  — ; 

R-  ^ 

r  is  imaginary  when  sin^(^  —  6')>  —- 

^1 


Art.  40.  —  Polar  Equations  of  the  Conic  Sectioxs 

Take  the  focus  as  pole,  the  axis  of  the  conic  section  as  polar 
axis.     From  the  definition  of  a  conic  section 


r  =  e-  DE  =  e  (DA  +  AE)  =e['-  +  r  cos  d 


Hence 


r—j^  +  ercofiO,  r 


1  —  e  cos  6 


74 


ANALYTIC  GEOMETRY 


Since  in  the  parabola  e  =  1,  the  polar  equation  of  the  parab- 

Ola  IS   r  =  :; 1,- 


In  the  ellipse  and  hyper- 
bola the  numerical  value  of 
the  semi-parameter  p  is 

a(l-e^); 
hence  the  polar  equation  of 
ellipse  and  hyperbola  is 

1  —  e  cos  0 

In  the  ellipse  e  is  less  than 
unity,    and     r    is     therefore 

Fig.  07.  .  -^ 

always   positive.     For   ^  =  0, 
r  —  a(l  +  e),  showing  that  the  pole  is  at  the  left-hand  focus. 
In  the  hyperbola  e  is  greater  than  unity,  and  r  is  positive 

from  6  =  0  to  6  =  cos~^  -  in  the  first  quadrant,  negative  from 

1  ^  1 

,-1^  ;r,   4-1.0   fi,.of  r.,no,iT.o,.f  fr>  fl  —  f-og-'-  in  the  fourth 


cos~'  -  in  the  first  quadrant  to 
e 


quadrant,  again  positive  from 
rant  to  ^  =  3G0' 


,^il 


cos^^  -  in  the  fourth  quad- 
e  1 

r  becomes  infinite  for  6  =  cos  ^  - ;  hence  lines 

'    1 


through  the  focus  making  angles  whose  cosine  is  -  with  the 

axis  of  the  hyperbola  are  parallel  to  the  asymptotes  of  the 
hyperbola. 

Problems. — 1.  The  length  of  the  perpendicular  from  the  pole  to  a 
straight  line  is  5  ;  this  perpendicular  makes  with  the  polar  axis  an  angle 
of  45°.     Find  the  equation  of  the  line  and  discuss  it. 

2.  Derive  and  discuss  the  polar  equation  of  the  straight  line  parallel 
to  the  polar  axis  and  8  above  it. 

3.  Derive  and  discuss  the  equation  of  the  straight  line  at  right  angles 
to  the  polar  axis,  and  intersecting  the  polar  axis  4  to  the  right  of  the  pole. 

4.  Derive  and   discuss   the   equation   of  the  circle,  radius  5,  center 

(10,     iTT). 


POLAR   COORDINATES 


75 


5.  Derive  and  discuss  Uie  equation  of  tlie  circle,  radius  10,  center 
(5,   Iw). 

6.  Derive  and  discuss  tlic  equation  of   the  circle,  radius  8,  center 
(10,    ]7r). 

7.  Derive  and  discuss  the  equation  of  the  circle 
(15,   tt). 

8.  Derive  and  discuss  the  equation  of  the  circle. 
(10,  ^tt). 

9.  Find  the  polar  equation  of  the  parabola  whose  parameter  is  12. 
Find  the  polar  equation  of  the  ellipse  whose  axes  are  8  and  6. 
Find  the  polar  equation  of  the  ellipse,  parameter  10,  eccentricity  \ 
Find  the  polar  equation  of  the  ellipse,  transverse  axis  10,  eccen 


radius  10,  center 


radius  10,  center 


10. 
11. 
12. 
tricity 
13. 
14. 
15. 


4  c2,-2  co^l , 


Find  the  polar  equation  of  the  hyperbola  whose  axes  arc  8  and  G. 

Find  polar  equation  of  hyperbola,  transverse  axis  12,  parameter  0. 

Find  polar  equation  of  hyperbola,  transverse  axis  8,  distance  be- 
tween foci  10. 

16.  Find  the  eijuation  of  the  locus  of  a  point  moving  in  such  a  manner 
that  the  product  of  the  distances 
of  the  point  from  two  fixed 
points  is  always  the  scjuare  of 
the  half  distance  between  the 
fixed  points.  This  curve  is  called 
the  lemniscate  of  Bernoulli. 

By  definition  ViV^  =  c^.    From 
the  figure  rx^ = r-  +  c^-2cr  cos  6, 

r^  =  r2  -f-  c2  +  2  cj-  cos  9,  hence  rrro-  —  r*  +  2  c^r'^  +  <: 
and  r2  =  2  ^2  (2  cos2  ^  -  1),  r^  =  2  c^  cos  (2  0). 

Corresponding  pairs  of  values  of  Vi 
and  r2  may  be  found  by  drawing  a 
circle  with  radius  r,  to  this  circle  a 
tangent  whose  length  is  c.  The  dis- 
tances from  the  end  of  the  tangent 
to  the  points  of  intersections  of  the 
straight  lines  through  the  end  of  the 
tangent  with  the  circumference  are 
corresponding  values  of  r^  and  j-o,  for 
7\S  ■  Tli  =  c'.  The  inter.sections  of 
arcs  struck  off  from  Fi  and  Fo  as 
centers  with  radii  TS  and  Tli  deter- 
mine points  of  the  lemniscate. 


76 


ANALYTIC  GEOMETRY 


17.  A  bar  turns  around  and  slides  on  a  fixed  pin  in  such  a  manner 
tliat  a  constant  lengtli  projects  beyond  a  fixed  straiglit  line.  Find  the 
equation  of  the  curve  traced  by  the  end  of  the  bar.  This  curve  is  called 
the  conchoid  of  Nicomedes. 


^^-^ 

Y 

-^^ 

^ ' 

—                       a, 

/ 

— 

m 

b 

A 

/ 

A 

X 

Take  the  fixed  point  A  as  focus,  the  line  ^X,  parallel  to  the  fixed  line 
mn,  as  polar  axis.  Call  the  distance  from  the  i^ole  to  the  fixed  line  h, 
the  constant  length  projecting  beyond  the  fixed  line  a.    Then 

The  conchoid  is  used  to  trisect  an  angle  graphically.  Let  GAH  be  the 
angle.  From  any  point  B  in  one 
side  of  the  angle  draw  a  perpen- 
dicular mn  to  the  other.  With 
vertex  of  angle  as  fixed  point, 
mn  as  fixed  line,  and  FG  =  2  BA 
as  constant  distance,  construct  a 
conchoid.  At  B  erect  perpen- 
dicular BC  to  mn,  and  join  its 
A  point   of    intersection   with    con- 

"'■  '  ■  choid  C  and  J.  by  a  straight  line. 

GAC  is  \  CtAII,  for,  drawing  through  D,  the  middle  point  of  BC,  a  parallel 
to  mn  and  joining  B  and  E,  the  triangles  ABE  and  BEC  are  isosceles. 
Kence  BAG  =  BE  A  =  2BCA  =  2  GAC. 


^ 

C      G 

-^ 

^^^ 

^^ 

^ 

\ 

D 
\ 

Ae 

"^ 

m 

\ 

n 

E 

F 

Art.  41. — Plottinct  of  Polar  Equations 

Example.  — Plot  r  =  10  cos  0. 

e  =  ()                 ^TT           ItT                fir                   TT                       fTT                §77             |  TT 

2,r 

r  =  l()     nV2       0       -5V2     -10     -5VL>       0    -  SV^ 

10 

POL  A  R    COORBINA  TES 


77 


If  tlie  iiuinl.or  oi' 
points  located  I'roiu  ^  =  0 
to  ^  =  2  TT  is  indefinitely 
increased,  the  polygon 
formed  by  joining  tlu? 
successive  points  ap- 
proaches the  circumfer- 
ence of  a  circle  as  its 
limit.  Tlie  form  of  the 
equation  shows  at  once 
that  the  locus  is  a  circle 
whose  radius  is  5. 


Example.  — Plot  r  =  aO. 

e  =  -4:         -3         -2 

-1 

0 

r  =  —  4a      —  3  a      —  2  a 

—  a 

0 

12        3        4 
a     2  a     3  a     4  a 


The  curve  is  called 
the  spiral  of  Archimedes. 
In  rectangular  coordi- 
nates the  equation  of 
this  spiral  is  transcen- 
dental. 


Example.  —  Plot  r  = 

$  =  0  cos->  f 

r  =  — 2  T  00 


3  —  T)  cos  6 


±cx> 


78 


ANALYTIC   GEOMETIIY 


From  6  =  0  to  6  =  cos~^  f ,  r  varies  continuovisly  from  —  2  to 

—  cc  ;  from  ^  =  cos~^f  to 
0  =  TT,  r  decreases  contimi- 
ously  from  +  co  to  -{-  -h  ; 
from  6  =  Tr  to  6  =  cos~^  |  in 
the  fonrtli  qxiadrant,  r  in- 
creases continuously  from  ^j 
to  +  CO  ;  from  6  =  cos~^f  in 
the  fourth  quadrant  to 
6  =  2  TT,    r    increases    from 

—  CO  to  —  2.  r  is  discon- 
tinuous for  ^=cos"'|.  This 
equation  represents  an  h}'- 
perbola    whose     less    focal 

distance  is  i,  greater  focal  distance  2,  semi-parameter  A,  eccen- 
tricity f 

Example.  —  Plot  /-^  =  8  cos  (2  9). 
0  =  0°  22.^°       45° 135°       157i°  180° 


r  =  ±  2.828      ±  2.378      0     imaginary     0      ±  2.37 


±  2.828 


From  180°  to  300°  the 
curve  is  traced  a  second 
time.  The  pole  is  a  cen- 
ter of  symmetry  of  the 
curve. ' 


Problems.  —  Plot 


COS  0 


cos  (6  —  J  tt) 

3.  r  =  ncnsi{3  0). 

4.  )•  =  2  cos  0. 

5.  r=  asm  (2  0). 


6.  r  =  acos(o0). 

7.  r  =  asin(Se). 

8.  »'  =  rtsin(4^). 

9.  r  =  a  sin  (5  6). 

10.  r  =  ^,  the  reciprocal  spiral. 

11.  r  —  a",  the  logaritlimic  spiral. 


POLAR   COOIiT)TNATES 


79 


12. 

,  the  lituus 

13. 

2 

'     1 

-  cos  0 

11 

5 

2 

-  3  cos  » 

15 

4 

'      3 

-  2  cos  » 

IB 

10 

1  +  cos  e 

17.  ?•  =  « (1  +  cos  e),  the  cardioid. 

18.  J- =  4(1  -COS0). 

19.  r  =  5  +  2  sin  0. 


20. 

r  —  2p  cot  ^  cosec  ff. 

21. 

^.  _      4  COS  ^ 

1  +  3  sin2  e 

22. 

4  COS  e 

1-5  sin-  0 

23. 

3  sin  e  cos  0 

sin^  d  +  cos-''  ^ 

24. 

r  =  rt(sin2e  +  cos2<?). 

25. 

r-  cos  (2  e)  -  4. 

26. 

)-2sin(2e)=8. 

27. 

r-  cos  1  e  =  2. 

28. 

)•-  =  10  sin  (2^). 

Art.  42. — Transformation  from  Rectangular  to  Polar 
Coordinates 

If  the  rectangular  equation  of  a  geometric  figure  is  given, 
and  the  polar  equation  is  desired,  find  the  values  of  the  rectan- 
gular coordinates  x  and  y  of  any  point  in  terms  of  the  polar 
coordinates  r  and  0  of  the  same  point;  substitute  in  the  rec- 
tangular equation  f(x,y)=0,  and  the  resulting  equation 
F(r,  6)=0  is  the  polar  equation  of  the  figure. 

Let  the  pole  A'  referred  to  the  rectangular  coordinates  be 
(m,  n),  6'  the  angle  made  by  the  polar  axis  with  the  X-axis. 
Then 

X  =  AD  =  m  +  r  cos  (6  +  6'), 

y  =  PD  =  n  +  r  sin  (d  +  6'). 

When  the  pole  is  at  the  origin, 
and  the  polar  axis  coincides  with 
the  X-axis,  these  formulas  be- 
come X  =  r  cos  6,  y  —  r  sin  $. 

If    the   polar   equation   of   a 
geometric   figure   is   given   and 
the  rectangular  equation  is  desired,   find  the  values  of   the 
polar  coordinates  r  and  6  of  any  point  in  terms  of  the  rectan- 


80  ANALYTIC  GEOMETRY 

gular  coordinates  x  and  y  of  the  same  point ;  substitute  in  the 
polar  equsition  F(r,  6)  =  0,  and  the  resulting  equation /(a;,  ?/)=:0 
is  the  rectangular  equation  of  the  curve. 
From  the  figure 

X  —  m 


r  =  y/{x  -  my  +{y-  w)^   cos  (6  +  6')  = 
sin(^  +  ^')-  •'~'" 


^{x  -  mf  +  (2/  -  ?0' 


V(.x'-m/+(y-n)^ 

When  the  origin  is  at  the  pole,  and  the  X-axis  coincides  with 
the  polar  axis,  these  formulas  become 

r  —  V.V-  +  ]j\   cos  Q  = ^  sin  9  = • 

VaT-  +  y-  Var  +  y- 

Problems.  — Transform  from  rectangular  coordinates  to  polar,  pole  at 
origin,  polar  axis  coinciding  with  A'-axis,  and  plot  the  locus  from  both 
equations. 

1.  x2  +  ?/^  =  25.  _  6.    2/2  =  i(4.T-a;2). 

2.  X-  +  2/2  -  10  X  =  0.  7.    ?/  =  -  1  (4  X  -  x^). 

3.  y^  =  2j-)X.  8.    if -Sx>j +  x^  =  0. 

4.  a;2  -  2/2  =  25.  9.    (x^  +  2/-)'^  =  «"  i^'  -  f')- 

5.  x2/  =  9. 

Transform  from  polar  coordinates  to  rectangular  coordinates,  X-axis 
coinciding  with  polar  axis. 

10.  r  =  «,  origin  at  pole.  ^^    ,.  ^ 9 ^  p^j^  ^^  ^^^  ^^ 

11.  r  =  10  cos  e,  origin  at  pole.  4  -  5  cos  ^ 

12.  9-2  =  «2  cos  (2  61) ,  origin  at  pole.       le.    r  = ,  pole  at  (4,  0). 

5  —  4cos& 

13.  )•-  cos .',  0  —  2,  origin  at  pole.  ,      cos  f 2  0^        .  •      .       , 

-  '       °  ^  17.    r-  =        '-      ',  origm  at  pole. 

„  cos*  e 

14.  ,.  = i^ ,  poleat(ip,  0).  ,      ..      n       ••      .      , 

1  -  cos  0  18.    r2cos'*^  =  1,  origni  at  pole. 


CHAPTER   VII 


PROPERTIES  OF  THE  STRAIGHT  LINE 


AuT.  43,  —  Equations  of  the  Straight  Line 

The  various  conditions  determining  a  straight  line  give  rise 
to  different  forms  of  the  equation  of  a  straight  line. 

I.   The  equation  of  the  straight  line  determined  by  tlie  two 
l.oints  (x',  ?/'),  (x",  y"). 

The  similarity  of  the  triangles 
rP'D  and  P'P"D'  is  the  geometric 
condition  which  locates  the  point 
P{x,  y)  on  the  straight  line  through 
P'ix',  y')  and  P"(.^•",  ?/")•  Tl>is 
condition  leads  to   the   equation 

y-y  ^ '^, — ^ 
x'  —  x" 

tancrular    coordinates 


^(x  —  o:').     In   rec- 


Fio.  77. 


tan  a,   where 


«     IS 


the   an,<rl( 


the 


line 


y  -?/  _ 
x'  —  x" 

makes   with    the    X-axis.     In    oblique   coordinate! 
sin  a 


-,  where  /3  is  the  angle  between  the  axes,  a 
x'  —  x"      sin  (/3  —  a) 

the  angle  the  line  makes  with  the  X-axis. 

II.  The  equation  of  a  straight  line  through  a  given  point 
(x',  y')  and  making  a  given  angle  «  with  the  X-axis  is 
y  —  y'  =  tan  «(a;  —  x').  If  the  point  {x',  y')  is  the  intersection 
(0,  n)  of  the  line  with  the  X-axis  and  tan  «  =  m,  the  equation 
becomes  y  =  7nx  +  n,  the  slope  equation  of  a  straight  line. 

On  the  straight  line  y  —  y'  —  tan  a{x  —  x')  the  coordinates  of 
the  point  whose  distance  from  (.«',  ?/')  is  (7,  are  x  =  x'  -f  d  cos  a, 
y  =  y'  -\-  d  sin  a. 

a  81 


82 


ANALYTIC  GEOMETRY 


III.    The  equation  of  the  straight  line  whose  intercepts  on 
the  axes  are  a  and  6. 

Let  {x,  y)  be  any  point  in  the 
line.     From  the  figure 


a—x_y 

a        b 

1,  the 


which  reduces  to  -  +  - 
a      h 
intercept  equation  of  a  straight 

Fig.  78.  Iji^g. 

IV.    When  the  length  p  and  the  inclination  «  to  the  X-axis 

of  the  perpendicular  from 
the  origin  to  the  straight 
line  are  given. 

Let  {x,  y)  be  any  point  in 
the  straight  line.  From  the 
figure,  AB+BC=p,  hence 

X  cos  (i  +  y  sin  «  =  p. 
This  is  the  normal  equa- 
tion of  a  straight  line. 
The  different  forms  of 
the  equation  of  a  straight  line  can  be  obtained  from  the  general 
first  degree  equation  in  two  variables  Ax  -\-  By  -\-  C  =  0,  which 
always  represents  a  straight  line. 

(a)  Suppose  the  two  points  (x',  y'),  (x",  y")  -to  lie  in  the  line 
represented  by  the  equation  ^x  +  i>?/ -|-C=  0.  The  elimina- 
tion of  A,  B,  C  from  (1)  Ax+By+C=0,  (2)  Ax'+By'+C=0, 
(3)  Ax"  -f  By"  +  C  =  0  by  subtracting  (2)  from  (1)  and  (3)  from 
(1),  and  dividing  the  resulting  equations  gives 


y-y  = 


y'  —  y' 

x'  -  x' 


(x-x'). 


(b)  Callin; 
X-axis  ((,  on 


:  the  intercept  of  the  line  Ax  -\-  By  +C  —0  on  the 

C 
the   F-axis  b,  for  y  =  0,  x  = -  =  a,  for  x  —  0, 


y  — —  b.     Substituting  in  the  equation  Ax  -f  By  +  C  =  0, 

there  results  --f-?^=l. 
a      h 


I'liOriCliTIES   OF  THE   STliAKniT  LINE  83 

(f)  Tlie  equation  .l.i-  +  B;/  +C=  U  may  be  written 

^       b'^    b' 

wliicli  is  of  the  form  y  =  vix  +  n. 

(d)  Let  Ax  +  Bij  +  C  =  0  and  x  cos  a  +  y  sin  a  =]>  repre- 
sent the  same  line.  There  must  exist  a  constant  factor  m 
such  that  VI Ax  +  vi  By  +  i>iC  =  0  and  x  cos  «  +  y  sin  «  —  p  =  0 
are  identical.  I'rom  tliis  identity  mA  =  cos  a,  mB  —  sin  a, 
-,itC  =  —2>-      The   iirst   two    equations   give   nrA^  +  vt^B'-  —  1, 

hence  iii  —  — -  That  is, 

Vvl-  +  B' 


^A'  +  B'        VA'  +  B'        y/A'  +  B' 
is  tlie  normal  form  of  the  e(]uation  of  the  straight  line  repre- 
sented by  Ax  +  By  +  C  =  0. 

The  nature  of  the  problem  generally  indicates  what  form  of 
the  equation  of  the  straight  line  it  is  expedient  to  use. 

Problems.  —  1.  AVrite  the  tuiuation  of  the  straight  line  through  the 
points  (2,3),  (-1,  4). 

2.  Write  tlie  eiiuation  of  the  straight  line  througli  (-  2,  3),  (0,  4). 

3.  Write  the  intercept  eijuation  of  tlie  straight  line  through  (4,  0), 
(0,  3). 

4.  Write  the  equation  of  the  straight  line  whose  perpendicular  dis- 
tance from  the  origin  is  5,  this  i)erpentlicular  malting  an  angle  of  30°  with 
the  A'-axis. 

5.  Write  the  e(iuation  ^  +  |  =  1  in  the  slope  form. 

6.  Write  the  equation  2  x  -  3  ?/  =  5  in  the  normal  form. 

7.  Write  the  equation  of  the  straight  line  through  (4,  -3),  making 
an  angle  of  135°  with  the  A'-axis. 

8.  On  the  straight  line  through  (-  2,  3),  making  an  angle  of  30°  with 
the  A'-axi-s,  find  the  coordinates  of  the  point  whose  distance  from 
(  -  2,  3)  is  0. 

9.  The  vertices  of  a  triangle  are  (3,  7),  (5,  -  1),  (-3,  5).  Write 
equations  of  meilians. 


84 


ANALYTIC   GEOMETRY 


Art.  44.  —  Angle  between  Two  Lines 

Let  V  be  the  angle  between  the  straight  lines  y  —  mx  +  n, 
y  =  m'x  +  n'.  From  the  figure 
V=  u  —  a',  hence 

.      Tr_   tan  a  —  tan  a' 
1  +  tan  a  tan  a' 
Since     tan  «  =  m,     tan  «'  =  7/1' 


—    tanF= 


When     the 


1  +  ?«?u' 
lines  are  parallel,  V=0,  which 
requires   that    m  —  m'.     When   the    lines   are   perpendicular, 

F=00°,  Avhich  requires  that  1  -\-mm'  =  0.  or  m' = 

m 
If   the    e<|uations    of    the    lines    are   written    in   the   form 

Ax  +  By+C=0,     A'x  +  ]^y+a  =  0,     tanF^^^-||. 

The   lines   are   parallel  when  A'B  —  AB'  =  0,    perpendicular 
when  AA'  +  BB'  =  0. 

The  equation  of  the  straight  line  through  (x',  y')  perpen- 
dicular to  y  =  mx  +  H  is  y  —  y'  = (x  —  x'). 

m 

The  equation  of  the  straight  line  through  (x',  y')  parallel  to 
y  =  mx  +  n  is  y  —  v'  —  ta  (x  —  x'). 

Let   the   straight   line    y  —  y'  =  tan  a'(x  —  x')    through    the 
point  (x',  ?/')  make  an  angle  6  with 
the    line    y  —  mx  -f  n.      From    the 
figure,  «'  =  ^  +  «.     Hence 
,        ,_  tan^+tan«  _  ta,n6  +  m 

1  —  tan  ^  tan  a  1—m  tan  6' 
since  tan  a  =  m.  Therefore  the 
equation  of  the  line  through  (x',  y') 


Fic.  81. 

making  an  angle  6  with  the  line  y  —  mx-\-n  is 
tan  6  +  m 
1  —  7)1  tan  ( 


y-y 


.(x-x'). 


PEOPEPiTIES   OF  THE  STliAiailT  LINE 
Problems. —  1.    Find   the   angle   the   line 


85 


•^  =  1   makes   with   the 
3 
A'-axis. 

2.  Find  the  angle  between  the  lines  2x  +  3y  =  1,  lx+  lij  =  1. 

3.  Find    the    e(iuation    of    the    line   through    (4,   -2)    parallel    to 
5x-7i/  =  10. 

4.  Find  the  equation  of  the  line  through  (1,  3)  parallel  to  the  line 
through  (2,  1),  (-3,  2). 

5.  Find  the  equation  of  the  line  through  the  origin  perpendicular  to 
3x-?/  =  5. 

6.  Find  the  equation  of  the  line  through  (2,  -  3)  perpendicular  to 
|a;-.\y  =  l. 

7.  Find  the  ecpation  of  the  line  through  (0,  -  5)  perpendicular  to 
the  line  through  (4,  5),  (2,  0). 

8.  The  vertices  of  a  triangle  are  (4,  0),  (5,  7),  (-0,  3).     Find  the 
equations  of  the  perpendiculars  from  the  vertices  to  the  opposite  sides. 

9.  The  vertices  of  a  triangle  are  (3,  5),  (7,  2),  (-  5,  -  4).     Find  the 
equations  of  the  perpendiculars  to  the  sides  at  their  middle  points. 

10.    Write  equation  of  line  through  (2,  5),  making  angle  of  45'  with 
2x-3i/  =  G. 

Akt.  45. —  Distance  from  a  Point  to  a  Line 

Write  the  equation  of  the  given  line  in  the  normal  form 
cccos«  +  ?/sin«-i:»  =  0.  Through  the  given  point  P{x',  y') 
draw  a  line  parallel  to  the 
given  line.  The  normal 
equation  of  this  xiarallel 
line  is 

X  cos  a  +  y  sill  (t  =  AP'. 

Since  (x',  y')  is  in  this  line, 

■  x'  cos  a  4-  y'  sin  «  =  AP'.  ^^^  ^, 

Subtracting  p  =  AD',  there 

results  x'cosa  +  y' sin  a- p=PD;  that  is,  the  perpendicular 

distance  from  the  point  {x',y')  to  the  line  x  cos  a  +  y  sin  a -p=0 


86  ANALYTIC   GEOMETRY 

is  x'  cos  «  +  y'  sin  «  —  p.  The  manner  of  obtaining  this  result 
shows  that  the  perpendicular  FD  is  positive  when  the  point  P 
and  the  origin  of  coordinates  lie  on  different  sides  of  the  given 
line ;  negative  when  the  point  P  and  the  origin  lie  on  the  same 
side  of  the  given  line. 

The  perpendicular  distance  from  {x\  y')  to  Ax  +  J5^  +  C'=  0 
is  found  by  writing  this  equation  in  the  normal  form 


V^'  +  B"        ^'A'  +  B"        V^-  +  B- 

j{x'  4-  Bii'  +  C 
and  api)lving  the  former  result  to  be   PD  =     '     „     '3= — 
i  L  ^     ^  V.l-  +  B' 

This  formula  determines  the  length  of  the  perpendicular;  the 
algebraic  sign  to  be  prefixed,  which  indicates  the  relative  posi- 
tions of  origin,  point,  and  line,  must  be  determined  as  before. 

Problems.  —  1.    Find  distance  from  (-2,3)  to  3  x  +  5  y  =  15. 

2.  Find  distance  from  origin  to  |  x  —  |  ?/  =  7. 

3.  Find  distance  from  (4,  -  5)  to  line  through  (2,  1),  (-3,  5). 

4.  Find  distance  from  (3,  7)  to  ^^_^  =  IjL=li^. 

5.  The  vertices  of  a  triangle  ar6  (3,2),   (-4,2),   (5,  -7).    Find 
lengths  of  perpendiculars  from  vertices  to  opposite  sides. 

6.  The  sides  of  a  triangle  are   ?/  =  2  x  +  5,  3  -  ^  =  !>  4  x  -  7  y  =  12. 
Find  lengths  of  perpendiculars  from  vertices  to  opposite  sides. 

7.  The  sides  of  a  triangle  are  2/  =  2x  +  3,  2/  =  -|x  +  2,  y  -x-b. 
Find  area  of  triangle. 


Art.  46.  —  Equations  of  Bisectors  of  Angles 

Let  the  sides  of  the  angles  be  Ax+By+C=0,  A'x+Bhj^-G'=0. 
The  bisector  ah  is  the  locus  of  all  points  equidistant  from  the 
given  lines  such  that  the  points  and  the  origin  lie  either  on  the 


I' HOP  Eli  TIES   OF  THE   STRAIGHT  LL\E 


87 


same  side  of  each  of  the  two  ; 
of  each  of  the  two  given  Hues. 
In  either  case  the  perpendicu- 
lars from  any  point  {x,  y)  uf 
the  bisector  to  the  given  lines 
have  the  same  sign,  and  the 
equation  of  the  bisector  is 
Ax  -f  Bif  +  C _  A'x  +  B'n  +  C" 


riven  lines  or  on  diiferent  sides 


V^-  +  B'  VA"  +  B" 

The  bisector  cd  is  the  locus 

of  all  points  equidistant  from  ^'"-  ^^• 

the  given  lines  and  situated  on  the  same  side  of  one  of  the 

given  lines  with  the  origin,  while  the  other  line  lies  between 

the  points  of  the  bisector  and  the  origin.     The  perpendiculars 

from  any  point  (x,  ?/)  of  the  bisector  cd  to  the  given  lines  are 

therefore  numerically  equal  but  with  opposite  signs,  and  the 

n ,,     T      ^        ,.    Ax  +  By  +  C         A'x  +  B'jf  +  C 
equation  of  the  bisector  m  is  .'  _ 


VA'  +  B' 


-VA"  +  B' 


Problems.  —  1.   Find  the  bisectors  of  the  angles  whose  sides  are 
3  .X  +  4  (/  =  5,  Hx  -  1  >j  =  2. 

2.  Find  the  bisectors  of  the  angles  whose  sides  are   ^x  —  ly  =  1, 
2/  =  2x-3. 

3.  Find  locus  of  all  points  cciuidistant  from  the  lines  2x  +  7  y  =  10, 
8  a;  —  5y  =  15. 

4.  The  sides  of  a  triangle  are  5a:  +  3  ?/  =  9,  l  x  +  I  y  =  I,  y  =  d x  -  10. 
Find  the  bisectors  of  the  angles. 

5.  The  sides  of  a  triangle  are  7x  +  5i/  =  14,  lOx  —  15y  =  21,  y  =  3x  +  7. 
Find  the  center  of  the  inscribed  circle. 


AuT.  47.  —  Lines  thuough  Ixtekskctiox  of  Givk.v  Lines 

Let  (1)  Ax  +  B>/+C.=  0  and  (2)  A'x  +  B'>/  +  C  =  0  Ije  the 
given  lines.  Then  (3)  Ax  +  By  +  C  -\-k (A'x  +  B'y  +  C)  =  0, 
where  k  is  an  arbitrary  constant,  represents  a  straight  line 


88  ANALYTIC   GEOMETRY 

tlirougli  the  point  of  intersection  of  (1)  and  (2).  For  equation 
(3)  is  of  tlie  first  degree,  hence  it  represents  a  straight  line. 
Equation  (3)  is  satistied  when  (1)  and  (2)  are  satisfied  simul- 
taneously, hence  the  line  represented  by  (3)  contains  the  point 
of  intersection  of  the  lines  represented  by  equations  (1)  and  (2). 
If  the  line  Ax  +  B>j+C  +  Jc (A'x  +  B'y  +  C")  =  0  is  to  contain 

the  point  (x\  y'),  k  becomes  -  ^,^,  _^  J^y  _^  ^r    Hence 

is  the  equation  of  the  line  through  (x',  y'),  and  the  intersection 
of  (1)  and  (2). 

If  the  equations  of  the  given  lines  are  written  in  the  normal 
form,  (1)  X  cos  a  +  ?/  sin  «  —  p  =  0,  (2)  x  cos  «'+  ?/  sin  «'  —  i''=  '*? 
the  A;  of  the  line  through  their  point  of  intersection 

(3)  X  cos  u  +  y  sin  a  —  2>  +  k  i^'  cos  a'  +  y  sin  a'  —  2>')  =  0 

-,.      ,  ^  ■    ■   i.  ^  i-         7  a;cos«+?/sin«— /> 

has  a  direct  geometric  interpretation.   A;  = — ■■ — -. ; :, 

a;cos« +?/sin«  — ^> 

that  is,  A;  is  the  negative  ratio  of  the  distances  from  any  point 
(x,  y)  of  the  line  (3)  to  the  lines  (1)  and  (2). 

Problems.  —  1.  Find  the  equation  of  the  line  through  the  origin  and 
the  point  of  intersection  of  3x  -  4?/  =  5  and  2  x  +  5?/ =  8. 

2.  Find  the  equation  of  the  locus  of  the  points  whose  distances  from 

the  lines  i;  x  -  5  y  +  2  =  0,  -  -  ?^  =  1  are  in  the  ratio  of  2  to  3. 
3      6 

3.  Find  the  equation  of  the  line  through  (-  2,  3)  and  the  intersection 
of  the  lines  8  x  -  5  ?/  =  15,  3  x  +  10  ?/  =  8. 


AiiT.   48.  —  TuuEE  Points  in  a  Straight  Line 

Let  the  three  points  (x',  ?/'),  (a-*",  y"),  {x"\  V'")  lie  in  a  straight 
line.     The  equation  of  the  straight  line  through  the  first  two 

points  is  y  -y'  =  -[,  ~-'„{x-x').      By  hypothesis  the  point 
x  —  X 


PnOPEliTlES   OF  THE  STRAIGUT  LINE  8'J 

?/'  —  ?/" 
(x'",  y'")  lies   in   tliis   liue,  hence  y"' —y' =  \,    —^,(^"' ~  ^')- 

Simplifying-,  (1)  x'y'"  -  x"y"'  +  x"y'  -  x"'y'  +  x'y"  -  x"'y"  =  0. 
When  this  equation  is  satisfied  the  three  points  lie  in  a 
straight  line,  whether  the  coordinates  are  rectangular  or 
oblique.  Notice  that  (1)  expresses  the  condition  that  the 
area  of  the  triangle  whose  vertices  are  (x',  y'),  (x",  y"), 
(x'",  y'")  is  zero. 

Problenls.  —  1.  In  a  parallelogram  each  of  the  two  sides  through  a  vertex 
is  prolonged  a  distance  equal  to  the  length  of  the  other  side.  Prove  that 
the  opposite  vertex  of  the  parallelogram  and  the  ends  of  the  produced  sides 
lie  in  a  straight  line. 

2.  In  a  jointed  parallelogram  on  two  sides  through  a  common  vertex 
two  points  are  taken  in  a  straight  line  with  the  opposite  vertex.  Show 
that  these  three  points  are  in  a  straight  line  however  the  parallelogram  is 
distorted. 

Art.  49. — Tiiiiek  Links  through  a  Point 

Let  the  three  lines  Ax  +  %  +  C'=  0,  A'x  +  B'y  +  C"  =  0, 
A"x-j- B"y -\- C"  =  0  pass  through  a  common  iwint.  IMako 
the  first  two  of  these  equations  simultaneous,  solve  for  x  and  y, 
and  substitute  the  values  found  in  the  third  equation.  There 
results 

AB'C"  +  A'B"C+A"BC'  -  A"B'C-A'BC"  -  AB"(J'  =  0, 
wliieh  is  the  condition  necessary  for  the  intersection  of  the 
given  lines. 

The  three  lines  necessarily  have  aconiiudu  ])oiut  if  constants 
K,,  K.,,  K;,  can  be  found  such  that  ki(Ax  -\-  By  +  C)+  k2{A'x  + 
J^'ll  +  C")+  K^{A"x  +  B"y  +  6'")=  0  is  identically  satisfied. 
For  the  values  of  x  and  y  which  satisfy  Ax  -\-  By  -\-  C  =0,  and 
A'x  -}-  B'y  -f-  C"  =  0  simultaneously  must  then  also  satisfy 
A"x  +  B"y  +  C"  =  0 ;  that  is,  the  point  of  intersection  of  the 
first  two  lines  lies  in  the  third  line. 

The  second  criterion  is  frequently  more  convenient  of  appli- 
cation than  the  first. 


90 


ANALYTIC   GEOMETRY 


Problems.  — 1.    The  bisectors  of  the  angles  of  a  triangle  pass  through 
a  common  point. 

Let  the  normal  equations  of  the  three  sides  of  the  triangle  be 
a;cosa  +  2/sina-i)i=0,  xcosj3  +  2/sin/3-i)2=0,  x  cos  7 +2/ sin  7-^93=0. 
Denote  the  left-hand  members  of  these 
\  equations  by  a,  /3,  7.     Then  0  =  0, 

|3  =  0,  7  =  0  represent  the  sides  of 
the  triangle,  and  a,  /3,  7  evaluated  for 
the  coordinates  of  any  point  (x,  y) 
are  the  distances  from  this  point  to 
the  sides  of  the  triangle.  Hence  the 
equations  of  the  bisectors  of  the  angles 
are  a-  p  =0,  ^-7=0,  7-0  =  0. 
The  sum  of  the  equations  of  the  bi- 
sectors is  identically  zero,  therefore 
the  bisectors  pass  through  a  common 
point. 

2.    The  medians  of  a  triangle  pass  through  a  common  point. 

For   every   point    in   the   median    through    C, 


hence 
Simi- 


sin  B  sin  A 
a  sin  A-  ^sinB  =  0  is  the  equation  of  the  median  through  C. 
larly  the  equation  of  the  median  through  B  is  found  to  be 

7  sin  C  —  a  sin  ^4  =  0; 
of  the  median  through  A,  /3  sin  B  -y  sin  C  =  0.     The  sum  of  these  equa- 
tions vanishes  identically. 


3.  The  perpendiculars  from  the  vertices  of  a  triangle  to  the  opposite 
sides  pass  tlirough  a  common  point. 

The  equation  of  the  perpendicular  through  C  is  aros  A  —  yScos  J5  =  0  ; 
through  B,  7  cos  C  -  a  cos  .1  =  0;  througli  A,  &  cos  27  -  7  cos  C  =  0. 


riiOPERTIES   OF   THE  STRAIGHT  LINE  91 


Akt.  50.  —  Tangent  to  Cukvk  ok  Skcond  Okdkr 

The  general  equation  of  the  curve  of  the  second  order  is 
ax- +  2bxy +  cy- -\-2dx  +  2e>/ +/=().  Let  (x„,  y^  be  any 
point  iu  the  curve.  The  equation  y  —  ?/„  =  tan  a{x  —  Xq)  repre- 
sents any  line  through  {xg,  y^.  The  line  cuts  the  curve  of  the 
second  order  in  two  points  and  is  a  tangent  when  the  two 
points  coincide.  The  coordinates  of  any  point  in  the  straight 
line  are  x  =  Xq-{-1  cos  a,  y  =  y^  -\- 1  sin  «.  The  points  of  inter- 
section of  straight  line  and  curve  of  second  order  are  the  points 
corresponding  to  the  values  of  I  satisfying  the  equation 

{ax,;  +  2  6.tv/o  +  cy,;  +  2  dx,  +  2  ey,,  +/) 
+  (2  a.rii  cos  «  +  2  hx.;^  sin  «  +  2  hy^  cos  « 
+  2  cyo  sin  «  +  2  r/  cos  «  +  2  e  sin  a)l 
+  (a  cos- «  -f  2  6  cos  «  sin  «  +  c  sin-  «)Z^  =  0. 

Since  (.i-q,  .Vo)  is  in  the  curve,  the  absolute  term  of  the  equation 
vanishes.  If  the  coefficient  of  the  first  poAver  of  I  also  van- 
ishes, the  equation  has  two  roots  equal  to  zero,  that  is  the  two 
points  of  intersection  6f  y  —  ?/„  =  tan  a{x  —  a-,,)  with  the  curve 
coincide  at  (xq,  ?/o)  when 

ax,^ cos  a  +  hx^  sin  «  +  hy^^  cos  «  +  ry,,  sin  a  -\-  d  cos  a-\-c  sin  a  =  0. 
The  equation  of  the  tangent  is  found  by  eliminating  cos  a  and 
sin«  from  the  three  equations  x  =  x^ -\- I  cor  a,  ?/  =  ?/„  + Z  sin  «, 
(ixq  cos  «  +■  bxo  sin  a  +  by„  cos  a  +  cy^  sin  a  +  fZ  cos  a-\-  e  sin  «  =  0. 
This  elimination  is  best  effected  by  multiplying  the  third 
equation  by  I,  then  substituting  from  the  first  two  equations, 
I  cos  a=x—Xo,  I  sin  a^y—y^.    The  resulting  equation  reduces  to 

axxo  +  b(xy,  +  x„y)  +  ryy,  +  r/(.r  +  .^•,.)  +  K.'/  +  ?/..)  +  /=  0. 
The  law  of  formation  of  the  equation  of  the  tangent  from  the 
equation  of  the  curve  is  manifest. 

Problems.  —  1.  Write  the  equation  of  tlie  tangent  to  x-  +  y"^  —  r"^  at 
(;^o,  yo). 


92  ANALYTIC  GEOMETRY 

2.  Write  the  equation  of  the  tangent  to  —  +  f^  =  1  at  (xo,  ?/o). 

d^      0- 

3.  Write  the  equation  of  the  tangent  to •'-  =  1  at  {xq,  yo). 

a-     b'^ 

4.  Write  the  equation  of  the  tangent  to  y-  =  2px  at  (xo,  yo). 

5.  Find  the  equation  of  the  tangents  to  4  x^  +  0  y-  =  30  at  the  points 
wliere  x~\. 

6.  At  what  point  of  x^  —  ?/2  =  1  nnist  a  tangent  be  drawn  to  make  an 
angle  of  45°  with  the  X-axis  ? 

7.  Find  the  angle  under  which  the  line  y  =  hx  —  b  cuts  the  circle 
x2  +  ?/2  =  49. 

8.  Find  tlie  angle  between  the  curves  y'^  =  C  a-,  9  2/2  +  4  ^2  =  30. 

9.  Find  the  equations  of  the  normals  to  the  ellipse,  hyperbola,  and 
parabola  at  the  point  {xo,  yo)  of  the  curve. 

The  normal  to  a  curve  at  any  point  is  the  perpendicular  through  the 
point  to  the  tangent  to  tlie  curve  at  the  point. 

10.  Wliere  must  the  normal  to  ^"  +  ^  =  1  be  drawn  to  make  an  angle 
of  135'"  with  the  X-axis  ? 

11.  Find  the  equation  of  the  normal  to  ?/2  =  10  x  at  (10,  10). 

12.  Find  equations  of  focal  tangents  to  ellipse  ^  +  f-  =  1. 

a2      b^ 


CHAPTER   VIII 

PKOPERTIES  or  THE  CIRCLE 
AuT.  51.  —  Equation  of  tiik  Cikclr 

The  equation  of  the  circle  referred  to  rectansjfuhir  axes, 
radius  E,  center  (a,  b),  is  (x  —  af  +{y  —  h)-  =  R'\  This  equa- 
tion represents  all  circles  in  the  Xl'-plane.  The  equation 
expanded  becomes  x^  -\-y~  —  2ax  —  2  by  -f  «-  +  ?>-  —  K'  =  0.  an 
equation  of  the  second  degree  lacking  the  term  in  .17/,  and  hav- 
ing the  coefficients  of  x^  and  y^  equal. 

Conversely,  every  second  degree  equation  lacking  the  term 
in  xy,  and  having  the  coefficients  of  x^  and  y-  equal,  represents 
a  circle  when  interpreted  in  rectangular  coordinates.  Such  an 
equation  has  the  form  x-  -f  ?/■  —  2  ax  —  2  by  +  c  =  0,  whicli 
when  written  in  the  form  (x  —  ay  -\-(y  —  b)'  —  (V  +  ^-  —  c,  is 
seen  to  represent  a  circle  of  radius  ((t- + /r' —  c)-,  with  center 
at  («,  b).  a,  b,c  are  called  the  parameters  of  tlie  circle,  aiul 
the  circle  is  spoken  of  as  the  circle  (a,  b,  0). 

When  the  center  is  at  the  origin,  a  =  0,  b  =^  0,  and  the  eipia- 
tion  of  the  circle  becomes  x^  +  y^=  R'- 

When  the  X-axis  is  a  diameter,  the  F-axis  a  tangent  at  the 
end  of  this  diameter,  the  circle  lying  on  the  ]>ositive  side  of 
the  F-axis,  a  =  R,  b  =  (),  and  the  ecjuation  of  the  circle  be- 
comes 2/^  =  2  Rx  —  x". 

Problems.  —  Write  tlie  cciuafions  of  tlic  fnllmvinn;  circles: 
1.    Center  (-2,  1),  radius  5.  2.    Center  (-  5,  5),  radins  5. 

3.    Center  (-  10,  15),  radius  5.  4.    Center  (0,  0),  radius  5. 

0:J 


94  ANALYTIC  GEOMETBY 

5.  Find  equation  of  circle  througli  (0,  0),  (4,  0),  (0,  4). 

6.  Find  center  and  radius  of  circle  through  (2,  —  1),  (—  2,  1),  (4,  5). 

7.  Find  center  and  radius  of  circle  x"^  -{■  y'^  -\-  i  x  —  Id  y  =  1 . 

8.  Find  center  and  radius  of  circle  x-  -\-  y-  +  l(i  x  =  11. 

9.  Does  the  line  3  x  —  5  y  =  12  intersect  the  circle 

a;2  +  j/2  -  8  X  +  10  2/  =  50  ? 
10.   Find  the  points  of  intersection  of  the  circles 

x2  +  2/2-10x  +  6y  =  20,    x2  +  2/2  +  4x-  15y  =  25. 


Art.  52.  —  Common  Chord  op  Two  Circles 

The  coordinates  of  the  points  of  intersection  of  the  circles 
a?  -{-y'^  -2ax  -2hy  -\-  c  =  Q,    x^ +  y^ -2a'x -2  b'y -\-c' =  0 

satisfy  the  equation 
(a;2  +  ?/2  -  2  ax  -  2  by  +  c)  -  (x^  +  y^-2  a'x  -  2  b'y  +  c')  =  0, 

which  reduces  to 

(a  -  a')x  +  (b  -  b')y  +(<■•  -  r)=  0. 

This  is  the  equation  of  the  straight  line  through  the  points  of 
intersection  of  the  circles,  that  is  the  equation  of  the  common 
chord  of  the  circles. 

The  intersections  of  two  circles  may  be  a  pair  of  real  points, 
distinct  or  coincident,  or  a  pair  of  conjugate  imaginary  points. 
Since  the  equation  of  the  straight  line  through  the  points  of 
intersection  is  in  all  cases  real,  it  follows  that  the  straight  line 
through  a  pair  of  conjugate  imaginary  points  is  real. 

Problems.  —  Write  the  equations  of  the  common  chords  of  the  pairs  of 
circles  : 

1.  X-  +  y-  -Gx  +  4y  =  12,     x"  +  y^  -  'ix  +  6y  =  12. 

2.  x2  +  ?y2  -  lOx  -  6?/=  15,     x^  +  ?/2  +  lOx  +  6?/ =  15. 

3.  x"  +  y-  +  lx  +  Sy  =  20,     x-  -y  y-  +  4x  -  \0y  =  18. 


PROI'EirriES   OF  THE  CIRCLE 


95 


AkT.    53. POWKR    OF    A    POIXT 

Let  (;c',  y')  be  any  point  in  the  plane  of  the  circle 

The  eqnation  of  any  straight  line  throngh  (x',  y')  is 

y  —  y'  =  tan  a  (x  —  x'), 
and  on  this  line  the  point  at  a  distance  d  from  (x',  y')  has  for 
coordinates   x  =  x'  +  d  cos  a,   yz=y'-\-d  sin  «.      The  distances 
from  (x',  y')  to  the  points  of  intersection  of  line  and  circle  are 
the  values  of  d  found  by  solving  the  equation 

|;(.^'  _  ay  +  iy'  -  by  -  R']  +  [2  (x'  -  a)  cos  a 
+  2(y'  -  h)  sin  «]  d  +  d-  =  0. 

Since  the  product  of  the  roots  of  an  equation  equals  numeri- 
cally the  absolute  term  of  the  equation,  it  follows  that  the 
product  of  the  distances  from  the  point  (x',  ?/')  to  the  points  of 
intersection  oi  y  —  y'  =  tan  «(x  —  x')  with  the  circle 

(.^  _  ay  +  (y  -  by  =  E"^  is    (x'  -  o)-  +  (y'  -  by  -  /?-. 

This  product  is  independent  of  a ;  that  is,  it  is  the  same  for 
all  lines  through  (x',  ?/').  This  constant  product  is  called  the 
power  of  the  point  (x',  ?/')  with  respect  to  the  circle. 

The  expression  {x'  —  a)--f  (?/'  —  by  —  Er  is  the  square  of  the 
distance  from  {x\  y')  to  the  center  (a,  b)  minus  the  square  of  the 
radius.  This  difference,  when  the 
point  (x',  ?/')  is  without  the  circle, 
is  the  square  of  the  tangent  from 
the  point  to  the  circle  ;  when  the 
point  (.!•',  )/')  is  within  the  circle, 
this  difference  is  the  square  of 
half  the  least  chord  through  the 
point. 

Let  *S'  represent  the  left-hand 
member  of  the  equation  xr  -\-  y-  —  2  a 
S  =  0  is  the  equation  of  the  circle,  and  S  evaluated  for  the  co- 


Fir,.  S". 

2  by  +  c  =  0.      Then 


96  ANALYTIC  GEOMETRY 

ordinates  of  any  point  (x,  y)  is  the  power  of  that  point  with 
respect  to  the  circle. 

Let  /i5i  =  0  and  So  =  0  represent  two  given  circles.  aS'i  =  S2 
is  the  equation  of  the  locus  of  the  points  whose  powers  with 
respect  to  /S'l  =  0  and  /S'2  =  0  are  equal.  This  equation,  Avhich 
may  be  written  Si  —  S-,  =  0,  represents  a  straight  line  called 
the  radical  axis  of  the  two  circles.  The  radical  axis  of  two 
circles  is  their  common  chord. 

If  three  circles  are  given,  Si  —  0,  S.,  =  0,  S-^=  0,  the  radi- 
cal axes  of  these  circles  taken  two  and  two  are  Si  —  S^—  0, 
S2  —  Ss  =  0,  S3  —  Si=:  0.  The  sum  of  these  three  equations  is 
identically  zero,  showing  that  the  radical  axes  of  three  circles 
taken  two  and  two  pass  through  a  common  point.  This  point 
is  called  the  radical  center  of  the  three  circles. 

Problems.  —  1.  Find  the  locus  of  the  points  from  which  tangents  to 
the  circles  x-  +  7/- -i-  4x  -  8y  =  b,  x^  +  2/2  -  6x  =  7  are  equal. 

2.  Find  the  point  from  which  tangents  drawn  to  the  three  circles 
a;2  +  y2  _  2  x  =  8,  x2  +  2/2  +  4  y  -  12,  x2  +  ?/-  +  4  x  +  8 ?/  =  5  are  equal. 

3.  Find  the  length  of  the  tangent  from  (—  3,  2)  to  the  circle 

(x  -  7)2  +  (y-  10)2  =  9. 

4.  Find  the  length  of  tlie  tangent  from  (10,  15)  to  the  circle 

x2-|- ?/2-4x +  C?/- 12. 

5.  Find  tlie  length  of  the  shortest  chord  of  the  circle 

x2  +  ?/2  -  0  X  +  4  ?/  =  3 
through  the  point  (—4,  3). 

6.  Find  the  equation  of  the  radical  axis  of  x2  +  ?/2  +  5.r  -  7 y  =  15, 
X-  +  2/2  -  3  a:  +  8  2/  =  10. 

7.  Find  the  radical  center  of  x-  +  y-  -  ".x  -  5,  xr  +  y"  -  4x  +  y  -  8, 
a;-  +  y^  +  7  2/  =  9. 

8.  Find  the  point  of  intersection  of  tlic  tln-ee  common  chords  of  the 
circles  x2  +  2/"  -  4x  -  2  2/  =  0,  x2  +  ?/2  +  2x  +  2  2/  =  ll,  x2-|- 2/"-6x  +  42/  =  17 
taken  in  pairs. 


I'JlorKin'lKS   OF   TIIK  CIliCLIC  97 


AuT.  54.  —  Coaxal  Svstkms 

Let  Si  —  0  and  iS.2  =  0  re[)i't'sent  two  circles.     Then 

,S',  -  kS,  =  0, 

for  all  values  of  the  parameter  A;,  represents  a  circle  tlirough 
the  intersections  of  Si  =  0,  aS^  =  0.  The  equation  ^i  —  kS.,  —  0, 
interpreted  geometrically,  gives  the  proposition,  the  locus  of  all 
the  points  vi^hose  powers  with  respect  to  two  circles  *Si  =  0, 
^2  =  0  are  in  a  constant  ratio  is  a  circle  through  the  points  of 
intersection  of  the  given  circles. 

Si  —  kS^  =  0,  by  assigning  to  k  all  possible  values,  represents 
the  entire  system  of  circles  such  that  the  radical  axis  of  any 
pair  of  circles  of  the  system  is  the  radical  axis  of  *S,  =  0  and 
S.,  =  0. 

If  tlie  parameters  of  *S'i  =  0  and  aS'2  =  0  are  a',  h',  r'  and 
a",  b",  c"  respectively,  the  parameters  of  Si  —  kSo=^  0  are 

a'  —  k(i"      h'  —  kh"      c'  —  kc" 
1-/0  '       1-k  '      1-k  ' 

Let  aS'  =  0  represent  a  circle,  L  =  0  a  straight  lino.  Then 
S  —kL  =  0  represents  the  system  of  circles  through  the  points 
of  intersection  of  circle  and  line.  The  commnii  radical  axis  of 
this  system  of  circles  is  the  line  L  =  0. 

Circles  having  a  common  radical  axis  are  called  a  coaxal 
system  of  circles. 

Problems. —  1.  Write  the  equation  of  tlie  system  of  circles  tlirougli 
the  points  of  intensection  of  x^  +  y-  —  2 x  +  G ?/  =  10  and  x-  +  y-  —  4y  =  8. 

2.  Find  the  equation  of  the  circle  through  the  points  of  intersection  of 
X-  +  y^-2x  +  Gy  =  0,  X-  ■+  y"^  -4y  =  8,  and  the  point  (4,  -  2). 

3.  Find  the  equation  of  the  circle  through  the  points  of  intersection  of 
oc^  +  2/2  4.  10  7/  =  6,   I X  -  1 2/  =  3,  and  the  point  (4,  5). 

4.  Find  the  equation  of  the  locus  of  all  the  points  which  have  etiual 
powers  with  respect  to  all  circles  of  the  coaxal  system  determined  by  the 
circles  x^  +  j/2  -  3x  +  7  y  =  15  and  x^  +  y'^  +  [>x  -  iy  =  12. 


98 


A  NA  L  YTIC  GEO  MET  R  Y 


Art.  55.  —  Okthogonal  Systems 
Two  circles 
a?  +  if  —  '^  a'x  ~-2b'y  +  c'  —  0,    x^  +  if  —  2  a"x  —  2  b"y  +  c"  =  0 

intersect  at  riglit  angles  when  the  square  of  the  distance  be- 
tween their  centers  equals  the 
sum  of  the  squares  of  their  radii ; 
that  is,  when 

{a'  —  a")-+{b'  —  b"f 

^a"+b'--c'  +  a"'+b"''-c", 
or    2a'a"  +  2b'b"  -c'  -c"  ^0. 

If  the  circle  (oj,  bi,  Cj)  cuts  each 
of    the    two    circles    (a',    b',    c'), 
(a",   b",   c")    orthogonally,    it   cuts   every    one   of   the  circles 

ka"     b'  -  kb"     c'  -  kc 


1-k         1-k         1-k 

of   the  coaxal  system  orthogonally.      For   the   hypothesis   is 
expressed  by  the  equations 

2a'a,  +  2  b%  -  c'-  c^  =  0,       2a"a,  +  2b"bi  -c"  -c,  =  0; 

the  conclusion  by  the  equation 


9  g'  —  ka"  cb'  —  ^'^'\ 

''l-k^'l-k     ' 


c'  -  kc' 
1-k 


0, 


which  is  a  direct  consequence  of  the  equations  of  the  hypothesis. 

The  condition  that  the  circle  (aj,  &i,  Cj)  cuts  the  circles 
(a',  b',  c')  and  («",  b",  c")  orthogonally,  is  expressed  by  two 
equations  between  the  three  parameters  ttj,  &„  Ci.  These  equa- 
tions have  an  infinite  number  of  solutions,  showing  that  an 
infinite  number  of  circles  can  be  drawn,  cutting  the  given 
circles  orthogonally. 

Let  Oi,  bi,  Ci  and  a.^,  bi,  c^  be  the  parameters  of  any  two  circles 
aSi  =  0,  aSj  =  0  cutting  aS'  =  0  and  S"  =  0  orthogonally.     Then 


PliOPEliTIES   OF  THE   CIRCLE 


99 


all  circles  of  the  coaxal  system  Si  -  kyS.,  =  0  cut  orthogonally 
all  circles  of  the  coaxal  system  S'  -  k'S"  =  0.  For  the  equa- 
tions 

2a'((i  +  -  f>'fJi  —  c'  —  c,  =  0, 

2  a"a,  + '2  b"h,  -  c"  -(\  =  0, 
2a'a.,  +  2b'b2-c'  -c.,  =  0, 
2  a"a.,  +  2  b"b2  -  c"  -  c.,=  0, 
have  as  consequence 
r,a'  —  k'a"  a,  —  k,cu  ,  r,b' —  k'b"  b^  —  k^b.2     c'  —  k 


(1) 
(2) 
(3) 
(4) 


Ci  —  kiC2_ 


1-k'      1-/h    ■"    1-k'     1-ki        1-k'        1-ki 

Subtracting  (2)  from  (1),  and  (3)  from  (1),  there  results 

2  (a'  -  a")  ai  +  2  (b'  -  b")  b,-c'  +  c"  =  0,  (5) 

2  (a,  -  a.)  a'  +  2  (6i  -  b.^  6'  -  Ci  +  C2  =  0.  (G) 

Equation  (5)  shows  that  the  centers  of  the  orthogonal  system 
Si  -  kySo  =  0  lie  in  the 
radical  axis  of  the  sys- 
tem S'—k'S"  =  0;  equa- 
tion (6)  shows  that  the 
centers  of  the  system 
S'  -  k'S"  =  0  lie  in  the 
radical  axis  of  the  sys- 
tem Si  -  kiS2  =  0. 

Take  for  X-axis  the 
line  of  centers  of  the 
system  S'  —  k'S",  for 
y-axis  the  radical  axis 
of  this  system.  Then 
the  equation  of  any 
circle  of  the  system 
becomes 
x-+y^-2a'x+c'  =  0.  (a) 

Since    by   hypothesis 


fiQ.  89. 


100  ANALYTIC   GEOMETRY 

tlie  power  of  (0,  ?/')  is  the  same  for  all  circles  of  the  system, 
c'  must  be  a  fixed  constant.  In  like  manner  it  is  found  that 
the  equations  of  the  orthogonal  system  Si  —  k^S^  —  0  have  the 

form 

ar+^?/^-2  6i^  +  Ci=0,  {(3) 

where  Cj  is  a  fixed  constant.  The  condition  for  the  orthogonal 
intersection  of  two  circles  when  applied  to  («)  and  (/3)  becomes 
Ci  =  —  c'.  Hence  the  equations  of  two  orthogonal  systems  of 
circles,  when  the  radical  axes  of  the  systems  are  taken  as 
reference  axes,  are 

X-  +  y^  —  2  a'x  +  c'  =  0,         x'^  +  11'—  l>'u  —  c'  =  0, 
where  «'  and  h'  are  parameters,  c'  a  fixed  constant. 

The  radii  of  the  circles  of  the  two  orthogonal  systems  are 
given  by  the  equations  r^  —  a^'^  —  c\  r'- =  h'" -\- c'  respectively. 
When  r  and  r'  become  zero  the  circles  become  points,  called 
the  point  circles  of  the  system.  In  every  case  one  of  the  or- 
thogonal systems  has  a  pair  of  real,  the  other  a  pair  of  imagi- 
nary, point  circles.* 

Problems.  —  1.  Find  the  equation  of  the  locus  of  the  centers  of  the 
circles  which  cut  orthogonally  the  circles  a;'-  +  y-  —  4  .x  +  0  y  =  15, 
x2  -I- 1/2  -f  5  X  -  8  2/  =  20. 

2.  Find  the  equation  of  the  circle  through  the  point  (2,  —  3)  and  cut- 
ting orthogonally  the  circles  x--\-y'^-]-^  x  —  7  2/  =  18,  x'^-l-y^  — 2  x—iy~\2. 

3.  Find  the  equation  of  the  circle  cutting  orthogonally  x'^+y'^  —  \0  x  =  9, 
3.2  4. 2/2  =  25,  x2  +  y-i-8y  =  IG. 

*  Through  every  point  of  the  plane  there  passes  one  circle  of  each  of 
the  orthogonal  systems.  The  point  in  the  plane  is  determined  by  giving 
the  two  circles  on  which  it  lies.  This  leads  to  a  system  of  bicircular 
coordinates. 

If  heat  enters  an  infinite  plane  disc  at  one  point  at  a  uniform  rate, 
and  leaves  the  disc  at  another  point  at  the  same  uniform  rate,  when  the 
temperature  conditions  of  the  disc  have  become  permanent,  the  lines  of 
equal  temperature,  the  isothermal  lines,  and  the  lines  of  flow  of  heat  are 
systems  of  orthogonal  circles.  The  points  where  the  heat  enters  and 
Jeaves  the  disc  are  the  point  circles  of  the  isothermal  system. 


rnOPEHTIKS  OF  THE  CIliC'LlS'  '^"^  "    11)1 

4.  Find  the  equation  of  the  system  of  circles  cutting  orthogonally  the 
coaxal  system  detennined  by  x'^-\-if+ix  +  6y-\5,  x^  +  y-+2  z-S  y  =  l2. 

5.  Write  the  equation  of  the  two  orthogonal  systems  of  circles  whose 
real  point  circles  are  (0,  4),  (0,  —  4). 


Art.  56.  —  Takgents  to  Circles 

The  equation  of  a  tangent  to  the  circle  x^  + -if  =  r'^  at  the 
point  (a*o,  ?/o)  of  the  circumference  is  xxq  +  yy^  =  r\ 

Let  (xi,  yi)  be  any  point  in  the  plane  of  the  circle  x--\-y'=i~, 
(x',  y'),  (x",  y"),  the  points  of  contact  of  tangents  from  (x^,  y^) 
to  the  circle.  Then  (.r,,  y{)  must  lie  in  each  of  the  lines 
xx'  +  yy'  =  7",  xx"  +  yy"  =  r-;  that  is,  x^x' +  y^y' =  r,  and 
Xix"  +  yiy"  =  rl  Hence  the  equation  of  the  chord  of  contact 
is  xxi  +  yyi  =  r^. 

The  distance  from  the  center  of  the  circle  to  the  chord  of 

contact  is -,  which  is  less  than,  equal  to,  or  greater 

(a.V  +  2/i-)' 
than  r,  according  as  the  point  (a-j,  y^  lies  without  the  circum- 
ference, on  the  circumference,  or  within  the  circumference.  In 
the  first  case  the  points  of  contact  of  the  tangents  from  (.Xi,  r/i) 
to  the  circle  are  real  and  distinct,  in  the  second  case  real  and 
coincident,  in  the  third  case  imaginary.  In  all  cases  the  chord 
of  contact  is  real. 

In  the  equation  y  =  mx  +  n  let  m  be  a  fixed  constant,  n  a 
parameter.  The  equation  represents  a  system  of  parallel 
straight  lines.  The  value  of  n  is  to  be  determined  so  that 
the  line  represented  by  y  =  mx  +  n  is  tangent  to  the  circle 
x^  +  y~  =  r^.  The  line  is  tangent  to  the  circle  when  the  per- 
pendicular from  the  center  of  the  circle  to  the  line  equals  the 
radius;  that  is,  when  =  7%  ?i=  ±rVl  +  m-.     There- 

fore,   the   equations   of    tangents   to    v?  ■{-  y'^  =  r  parallel   to 
y  =  mx  -f  n  are  y  =  mx  ±  r  Vl  +  m'. 


102  ANALYTIC  GEOMETRY 

Problems.  —  1.   Find  the  equations  of  the  tangents  to  x-  -\-  y-  —  25  at 
x  =  3. 

2.  Find  the  chord  of  contact  of  tangents  from  (2,  —3)  to  x-  +  y"^  =  1. 

3.  Find  the  points  of  contact  of  tangents  from  (5,  7)  to  x-  +  y-  —  9. 

4.  Find  the  equations  of  tangents  to  x-  +  ?/-  =  16,  making  angles  of 
45°  with  the  A'"-axis. 

5.  Find  the  equations  of  tangents  to  x^ +?/'-  =  25  parallel  to  ?/=3  x+5. 

6.  Find  the  equations  of  tangents  to  x^  +  y'-  =  25  perpendicular  to 
2/  =  3  X  +  5. 

7.  Find  the  slopes  of  tangents  to  x-  +  ?/-  =  9  through  (4,  5). 

8.  Find  the  equations  of  the  tangents  to  x^  +  ?/2  =  \Q  through  (5,  7). 

9.  The  chord  of  contact  of  a  pair  of  tangents  to  x'  +  ?/2  =  25  is 
2  X  +  3  //  =  5.     Find  the  intersection  of  the  tangents. 

10.    Find  equation  of  tangent  to  (x  —  a)^  +  (y  —  b)-  —  f-  at  (x',  y')  of 
circumference. 

Art.  57.  —  Poles  and  Polars 

Since  it  is  awkward  to  speak  of  the  chord  of  contact  or  the 
point  of  intersection  of  a  pair  of  imaginary  tangents,  the  point 
(xi,  ?/i)  is  called  the  pole  of  the  straight  line  xx^  +  yi/i  —  i^  with 
respect  to  the  circle  s?  -\- y-  =  r,  and  xx^  +  yy^  —  r^  is  called  the 
polar  of  the  point  (xj,  y^).  (x^,  ?/,), 
which  may  be  any  point  of  the 
plane,  determines  uniquely  the 
line  xxi  +  yyi  —  r- ;  and  conversely, 
xxi  -\-  yyi  =  r,  which  may  be  any 
straight  line  of  the  plane,  deter- 
mines uniquely  the  point  (iCj,  y^. 
The  relation  between  pole  and  polar 
therefore  establishes  a  one-to-one  correspondence  between  the 
points  of  the  plane  and  the  straight  lines  of  the  plane. 

The  polar  of  (.rj,  ?/j)  with  respect  to  ar-|- ?/-=?-  is  xxi+yy^=r-, 

the  line  through  the  center  of  the  circle  and  {x^,  ?/i)  is  ?/  =  ' '.t". 

Hence  the  line  through  the  pole  and  the   center   is   perpen- 


PliOrEllTIES   OF  THE   (JIRCLK 


103 


dicular  to  the  polar,  and  the  angle  included  by  lines  fiom  the 
center  to  any  two  points  equals  the  ant,de  included  l)y  the 
polars  of  the  two  points. 

The  distance  from  the  center  of  the  circle  .r  +//-  =  r  to  the 


l)olar  of  (.1-,,  ?/i)  is 


that  is,  the  radius  is  the  geomet- 


ric  mean  between  the  distances  of  the  center  from  pole  and 
polar. 

The  i)olar  of  (.).•„  y^),  with  respect  to  the  circle  x-  +  U'  =  t",  is 
constructed  geometrically  by  draAving  a  perpendicular  to  the 
line  joining  (.t„  ?/i)  and  the  center  of  the  circle  at  the  point 
whose  distance  from  the  center  is  the  third  proportional  to  the 
distance  from  (a*,,  ?/,)  to  the  center  and  the  radius  of  the  circle. 
The  pole  of  any  line,  with  respect  to  the  circle  x-  -\-  y-  =  r,  is 
constructed  geometrically  by  laying  off  from  the  center  on  the 
perpendicular  from  the  center  to  the  line  the  third  propor- 
tional to  distance  from  center  to  line  and  the  radius  of  the 
circle. 

The  polar  of  (.r„  y{)  Avith  respect  to  the  circle  x- +  y-  =  r-  is 
xxi  +  yyi  =  ?•-,  the  polar  of  (x.,,  y.,)  is  xx^  +  yy-j  =  r-  The  condi- 
tion which  causes  (a'l,?/,) 
to  lie  in  the  polar  of 
(X2,y2)is  x^Xo+yiy2^r^\ 
this  is  also  the  condi- 
tion which  causes  the 
polar  of  (.r„  //i)  to  con- 
tain (x.,,  y-_^.  Hence 
the  polars  of  all  points 
in  a  straight  line  pass 
through  the  pole  of  the 
line,  and  the  poles  of 
all  lines  through  a 
point  lie  in  the  polar 
of  that  point. 


104  ANALYTIC  GEOMETRY 

Problems.  —  1.    Write  the  equation  of  the  polar  of  (2,  :])  with  respect 
to  x~  +  y-  =  10. 

2.  Find  the  point  whose  polar  with  respect  to  x-  +  y-  =  d  is  S  x  +  7  y  = 

18. 

3.  Find  distance  from  center  of  circle  x-  +  y-  =  26  to  polar  of  (3,  4). 

4.  Find  equation  of  polar  of  {x',  y')  with  respect  to  circle  (x  -  a)'^ 
+  (2/  -  '0-  =  '•-• 

5.  Find  polar  of  (0,  0)  with  respect  to  x-  +  y-  =  r^. 


Art.  58.  —  Reciprocal  Figures 

If  a  geometric  figure  is  generated  by  the  continuous  motion 
of  a  point,  the  polar  of  the  generating  point  takes  consecutive 
positions  enveloping  a  geometric  figure.  To  every  point  in  the 
first  figure  there  corresponds  a  tangent  to  the  second  figure ; 
to  points  of  the  first  figure  in  a  straight  line  there  correspond 
tangents  to  the  second  figure  through  a  point;  to  a  multiple 
point  of  the  first  figure  there  corresponds  a  multiple  tangent 
in  the  second.  If  two  points  of  intersection  of  a  secant  of 
the  first  figure  become  coincident,  in  which  case  the  secant 
becomes  a  tangent,  the  pole  of  the  secant  at  the  same  time 
must  become  the  point  of  intersection  of  two  consecutive  tan- 
gents of  the  second  figure,  that  is  a  point  of  the  second  figure. 
Hence  the  first  figure  is  also  the  envelope  of  the  polars  of  the 
points  of  the  second  figure.  For  this  reason  these  figures  are 
called  reciprocal  figures.  Reciprocation  leads  to  the  principle 
of  duality  in  geometry.* 

Problems.  —  1.  To  find  the  reciprocal  of  the  circle  C  with  respect  to 
the  circle  0,  x-  +  y-  =  r^. 

*  The  principle  of  duality  was  developed  by  Poncelet  (1822)  and  Ger- 
gonne  (1817-18)  as  a  consequence  of  reciprocation,  independently  of 
reciprocation  by  Mobius  and  Gergonne, 


pnoPEirriEs  of  the  circle 


105 


The  line   nir{xxx  +  mn  =  f-)    is   tlie   polar   of    the  center   C{xu  2/i) 
with  respect  to  the  circle  0  ;  p  (Xa,  2/2)  is  the  pole  of  any  tangent 

PT{xx.z  +  ijih  =  r^) 
to  the  circle  C.     Then 

OC  =  (:'•!- +  2/1-)', 


pK 


CP 


X1X2  +  yiyo  -  r- 


X\Xi  + 


Op^(x2-+y2~)l 
Hence 

OC-pK=  CP-  Op, 

or         ^  =  ^'. 
pK      CP 

oc 

—^  is  constant,  and  therefore  p  nmst  generate  a  conic  section  whose  focus 

00 

is  O,  directrix  ////',  eccentricity  -— .     This  conic  section   is  an  ellipse 

when  O  is  within  the  circumference  of  the  circle  C,  a  parabola  when  0  is 
on  the  circumference,  an  hyperbola  when  0  is  without  the  circumference. 

2.   Find  the  reciprocal  of  a  given  triangle. 

Call  the  vertices  of  the  given  triangle  A{xx,  y{),  B{xn,  2/2),  C(xz,  yz)- 
The  polars  of  these  vertices  with  respect  to  x^  +  y~  =  r'^  are 

bc(xxx  +  2/2/1  =  '•^)' 

ac{xx2  +  2/2/2  =  »•-), 

a5(a:a:3  +  2/2/3  =  »•')• 
Triangles  such  that  the  ver- 
tices of  the  one  are  the 
poles  of  the  sides  of  the 
other  are  called  conjugate 
triangles.  The  conjugate 
triangle  of  the  triangle  cir- 
cumscribed about  a  circle 
with  respect  to  that  circle 
is  the  triangle  formed  by 
joining  the  points  of  con- 
tact. 


106  ANALYTIC   GEOMETRY 

3.  The  straight  lines  joining  the  corresponding  vertices  of  a  pair  of 
conjugate  triangles  intersect  in  a  common  point. 

The  equations  of  the  lines  through  the  corresponding  vertices  are 
Aa,  (rciX3  +  2/12/3  -  r'^)  {xx^  +  yy^  -  r~) 

-  (a;iX2  +  2/12/2  -  r'^){xxz  +  2/2/3  -  r-)  =  0  ; 
Bh,  {xiXo  +  2/12/2  -  r^){xxz  +  2/2/3  -  r-) 

-  {x^xz  +  2/22/3  -  r^)  {xx^  +  2/^1  -  V'-)  =  0  ; 
Cc,  {XiXs  +  yzys  -  r^)  (xxi  +  2/2/1  -  r^) 

-(X1X3  +  2/12/3  -  r'^)  {xx2  +  2/2/2  -  »■")  =  0. 
The  sum  of  these  equations  is  identically  zero,  therefore  the  lines  Aa, 
Bb,  Cc,  pass  through  a  common  point. 

4.  Show  that  if  a  triangle  is  circumscribed  about  a  circle  the  straight 
lines  joining  the  vertices  with  the  points  of  contact  of  the  opposite  sides 
pass  through  a  common  point. 

5.  Reciprocate  problem  3. 

The  figure  formed  by  the  conjugate  triangles  ABC,  ahc  is  its  own 
reciprocal.  The  poles  of  the  lines  joining  the  corresponding  vertices  of 
ABC  and  abc  are  the  points  of  intersection  of  the  corresponding  sides 
of  ABC  and  abc.  Hence  the  reciprocal  of  problem  3  is,  the  points  of 
intersection  of  the  corresponding  sides  of  a  pair  of  conjugate  triangles  lie 
in  a  straight  line. 

6.  Reciprocate  problem  4. 

The  reciprocal  of  the  circle  is  a  conic  section,  the  reciprocals  of  the 
points  of  contact  of  the  sides  of  the  triangle  are  tangents  of  the  conic  sec- 
tion, the  reciprocals  of  the  vertices  of  the  triangle  are  the  chords  of  the 
conic  section  joining  the  points  of  tangency,  hence  the  poles  of  the  lines 
from  the  vertices  to  the  points  of  contact  of  the  opposite  sides  in  the  given 
figure  are  the  points  of  intersection  of  the  sides  of  the  triangle  inscribed 
in  the  conic  section  with  the  tangents  to  the  conic  section  at  the  opposite 
vertices  of  the  triangle.  These  three  points  of  intersection  must  lie  in  a 
straight  line. 

Art.  59.  —  Inversion* 

Let  P'(.T„  ?/i)  be  any  point  in  the  plane  of  the  circle  x-  +  y-=r", 
P(x,  11)  the  intersection  of  the  polar  of  P',  (1)  xx^  +  ?///i  =  i~,  and 

*  The  value  of  inversion  in  geometric  investigation  was  shown  by 
Pliicker  in  1831.  The  value  of  inversion  in  the  theory  of  potential  was 
shown  by  Lord  Kelvin  in  1845. 


PUOPERTIES   OF  THE  CIRCLE 


107 


the  diameter  through  P',  (2)  y  =  -Av.     Then  OP  •  OP'  =  r'-,  that 


^Vheu  r  becomes 


is  (r  -  PA)(r  +  P'A)=  r-,  ^vheiice 

J 1_^1 

PA     FA     V 

infinite  the  circle  becomes  a 
straight  line,  PA  =  P'A,  and  P 
and  P'  become  symmetrical  points 
with  respect  to  the  line.  P  is  said 
to  be  obtained  from  P'  by  inver- 
sion,   by    the    transformation    by 

reciprocal  radii  vectors,  or  by  symmetry  with  res})ect  to  the 
circle.  This  transformation  establishes  a  one-to-one  corre- 
spondence between  the  points  within  the  circle  and  the  points 
without  the  circle. 

The  coordinates  of  P  are  obtained  in  terms  of  the  coordinates 
of  P'  by  making  (1)  and  (2)  simultaneous  and  solving  for  x 

and  ?/.     There  results  x  =  — — -'— y,  y  =  -—-—• 
xl  +  y,-  Xi-  -\-  y^ 


ilarly. 


t^x 


2/1 


r'y 


X'  +  y-  X-  +  y- 

If  the  point  (.t„  ?/,)  describes  a  circle 

x,\-\-y:--2aj\-2hy,  +  c  =  0, 
the  inverse  point  (.r,  ?/)  traces  a  curve  whose  equation  is 
r\-c-  4-  r'^v"       2  arx       2  hr>i 


(x-  +  iff      x^  +  y-     X-  +  y- 
which  reduces  to 


+  c  =  0, 


o  ,     o      sar- 

^■"  +  v ;« 

c 


2  hr 


y  + 


the  erpiation  of  a,  circle.     Hence  iuvcrsiou  Iransrorms  llie  circle 


(a,  b,  c)  into  the  circle 


("alliu''-  the  radius  of  tlie 


108 


ANALYTIC  GEOMETRY 


given  circle  R,  the  radius  of  the  transformed  circle  R', 


R-  =  ^+^J^. 


That  is, 


c.        c- 


R  =  -R. 


When  c  =  0,  R'  =  cc;  that  is,  the  transformed  circle  becomes 
a  straight  line,  c  =  0  is  the  condition  which  causes  the  center 
of  the  inversion  circle,  which  has  been  taken  at  the  origin  of  co- 
ordinates, to  lie  in  the  circumference  of  the  given  circle  (a,  6,  c). 
The  inverse  of  a  geometric  figure  may  be  constructed  mechan- 
ically by  means  of  an  apparatus  called  Peaucellier's  inversor. 

The  apparatus  consists  of  six 
rods,  four  of  equal  length  h 
forming  a  rhombus,  and  two 
others  of  equal  length  a  con- 
necting diagonally  opposite 
vertices  of  the  rhombus  with 
a  lixed  point  0.  The  rods 
are    fastened    together    by 

Fig.  95.  .  i.         n  r      4. 

pins  so  as  to  allow  perfect 
freedom  of  rotation  about  the  pins.  If  P  is  made  to  follow  a 
given  curve,  P'  traces  the  inverse,  the  center  of  inversion  being 
O  and  the  radius  of  inversion  (p?  -f  If)^.     For 

OP=a cos 6  —  bcos  6',  OP  =^acos6  +  b  cos  6',  a  sin 0=1  sin 6>'. 

Hence  OP  •  OP'  =  a-  cos-  d  -  Ir  cos^  6',  o?  sin-  6  -h-  sin-  6'  =  0, 

and  by  addition  OP  •  OP  =  cr  -  Jr. 

If  the  point  P  describes  the  circumference  of  a  circle  passing 
through  0,  P  must  move  in  a  straight  line.  Therefore  the 
inversor  transforms  the  circular  motion  of  P  about  0',  mid- 
way between  0  and  P,  as  center,  into  the  rectilinear  motion 
of  P'. 


rUorKllTlKS    OF  THE   CIRCLE 


109 


The  cosine  of  the  angle  between  two  circles  {a,  h,  c),  (a',  h',  c') 
is  found  from  the  ecjuation 

(rt  _  a'f  +  {b  -  b'f  =  r-  +  r"  -  2  rr'  cos  6 

.  2aa'  +  2  hh'  -  c  —  d 


to  be 


2Va^  +7>2  -  c  V^2  ^  in  _  ^ 


The  circles  obtained  by  inverting  the  given  circles  are 
Calling  their  included  angle  $', 


cos  ^' 


2ffa'>-''     2  ^j?/r^  _  ?;;  _  r^ 
cc'  cc'  c      c' 


which  reduces  to 

cos  9'  = 


2  «a'  +  2  6/y  -  c  -  c' 


2  Va^  +  ?/  _  c  V«''  +  6'2  -  c' 


Hence  the  angle  between  two  circles  is  not  altered  by  inversion. 
For  this  reason  inversion  is  called  an  equiangular  or  conformal 
transformation. 

If  two  orthogonal  systems  of  circles  are  inverted,  taking  for 
center  of  inversion  one  of  the  points  of  intersection  of  that 


110  ANALYTIC  GEOMETRY 

system  of  circles  which  has  real  points  of  intersection,  one  of 
the  systems  of  circles  transforms  into  a  system  of  straight  lines 
through  a  point.  Hence  the  other  system  of  circles  must  trans- 
form into  a  system  of  concentric  circles  whose  common  center 
is  this  point. 


CHAPTER   IX 


PROPERTIES  OF  THE  CONIO  SECTIONS 


Akt.  60.  —  General,  Equation 

A  point  governed  in  its  motion  by  the  law  —  the  ratio  of 
the  distances  from  the  moving  point  to  a  fixed  point  and 
to  a  fixed  line  is  constant  —  generates  a  conic  section.  To 
express  this  definition  by  an  eqnation  between  the  coordinates 
of  the  moving  point,  let  the  moving  point  be  (x,  y),  the  fixed 
point  F,  the  focus  (in,  n),  the  fixed  line  UH'.  the  directrix 
a; cos  a  +  y  sin  «  —  ^^  —  0.  Calling 
the  constant  ratio  e,  the  defini- 
tion is  expressed  by  the  equation 
PF'  =  e^  •  PD^,  which  becomes 

{m  -  xf  +  (n  -  yf 

=  e-  (x  cos  «  4-  2/  sin  a  —  py. 

a  is  the  angle  which  the  axis  of 
the  conic  section  makes  with  the 
X-axis,  1^  the  distance  from  the 
origin  to  the  directrix. 

By  assigning  to  m,  n,  e,  a,  p  their  proper  values  in  any 
special  case,  this  general  eqnation  becomes  the  equation  of 
any  conic  section  in  any  position  whatever  in  the  XF-plane. 
For  example,  to  obtain  the  common  equation  of  the  ellipse, 
which  is  the  equation  of  the  ellipse  referred  to  its  axes,  make 

m  =  ae,  n  =  0,  a  —  0,  p  =  "^,  1  —  e^  =  ';,•     The  general  equation 


a   -, 
e 


112 


A NA L  YTIC  GEOMETR  Y 


becomes  (ae  —  .^•)-  -\- y-  =  {ex  —  a)-.     Expanding  and  collecting 
terms,  /  +  (1  —  e-)  X'  =  a-  (1  —  e-),  or  ^  +  ^  =  1. 

To  obtain  the  equation  of  the  hyperbola  referred  to  its  axis 
and  the  tangent  at  the  left-hand  vertex,  make  m  —  a(l  +  e), 


„  =  0,  «  =  0,   p='^Sl+-^,  l-e'  =  ---     The  general  eqna- 

e  a- 

tion   becomes    (a  +  ae  —  a-)-  +  f  =  (ex  -  a  —  ae)-.      Expanding 
and  collecting  terms,  ?/-  =  (1  —  e-)  (2  ax  —  x"),  or 


?/" 


^4(2rtx--x-). 


Problems.  — From  the  general  equation  of  a  conic  section  referred  to 
rectangular  axes,  obtain : 

1.  The  common  equation  of  the  hyperbola. 

2.  The  common  equation  of  the  parabola. 

3.  The  equation  of  the  ellipse  referred  to  its  axis  and  the  tangent  at 
the  left-hand  vertex. 

4.  The  equation  of  the  ellipse  referred  to  its  axis  and  the  tangent  at 
the  right-hand  vertex. 

5.  The  equation  of  the  hyperbola  referred  to  its  axis  and  the  tangent 
at  the  right-hand  vertex. 

6.  The  equation  of  the  parabola  referred  to  its  axis  and  the  perpen- 
dicular to  the  axis  through  the  focus. 

7.  The  equation  of  the  ellipse  referred  to  its  axis  and  the  perpendicu- 
lar to  the  axis  through  the  focus. 

8.  The  equation  of  the  hyperbola  referred  to  its  axis  and  the  directrix. 


PUOPEUTIES   OF  THE  CONIC  SECTIONS 


113 


9.    'I'hc  c'liuation  of  tlir  panibola  referred  to  its  axis  and  the  dirrctrix. 

10.  Show  that  in  the  hyperbolas  ^"-'^"=1,  ■'/' _  ^  =  i  tlie  traiis- 

a-      b'^  b-      u- 

verse  axis  of  the  first  is  the  conjugate  axis  of  tlie  second,  and  vice  versa. 
Such  hyperbolas  are  called  a  pair  of  conjugate  hyperbolas. 

11.  Derive  from  the  general  equation  of  a  conic  section  the  equation 


fc2 


of  the  hyperbola  conjugate  to  — 

m  =  0,  n  -  be,  a  =  90^  p  = '\  1  -  e^  =  -  ?^ 
e  b- 

12.  Show  that  the  straight  lines  tj  =±-x  are  the  conunon  asymptotes 

a 

of  the  pair  of  conjugate  hyperbolas  —  —  ''-  =  1,  —  —  •''  =  —  1. 
«-      b-  a-     b- 

13.  Find  the  equation  of  the  ellipse  focus  (—3,2),  eccentricity  |, 
major  axis  10,  the  axis  of  the  ellipse  making  an  angle  of  45°  with  the 
A'-axis. 

14.  Find  the  equation  of  the  ellipse  whose  focal  distances  are  2  and  8, 
center  (5,  7),  axes  parallel  to  axes  of  reference. 

15.  Find  the  equation  of  the  hyperbola  whose  axes  are  10  and  8,  cen- 
ter (3,  —  2),  axis  of  curve  parallel  to  X-axis. 

16.  Find  the  equation  of  the  parabola  whose  parameter  is  0,  vertex 
(2,  —  3),  axis  of  parabola  parallel  to  A'-axis. 


Art.    61.  —  Tangents  and  Nokmals 


Using  the  common  equations  of  ellipse,  hyperbola,  and  parab- 
ola, the  equations  of   tangents  to  these  curves  at  the  point 

(.Tu,   y/o)    of   the    curve    are 

tt- "^  V  '  a-  lr~  ' 
yjhi=2){x-\-x^,  respectively. 
The  slopes  of  these  tangents 

are  for  the  ellipse 


for  the  hyperbola  — ■'-,  for 


114 


ANALVriC  GEOMETRY 


the  parabola  —     Calling  the  intercepts  of  the  tangent  on  the 

1ft)  2  7  2 

X-axis  X,  on  the  F-axis  Y,  for  the  ellipse  X  =  —,  Y=—,  for 
the   hyperbola  X  =  ^,    Y= ,  for  the  parabola  X  —  —  Xo, 

y=  i_?/||.     X  and  y  may  in  each  case  be  determined  geometri- 
cally, and  the  tangent  drawn  as  indicated  in  the  figure. 


Suppose  the  point  (,<■„  ?/,)  to  be  any  point  in  the  plane  of  the 
ellipse  "-T,  +  •—  =  1.  Let  {x',  y'),  (x",  y")  be  the  points  of  con- 
tact of  tangents  from  (a-j,  y{)  to  the  ellipse.     Then  must  (xi,  y^) 

...  ,      „    ,     ,.         xx'      ?/?/'      .     xx"      ?/w"     ^    ,,    ,  .     ,, 

he  m  each  of  the  hues  -^  -f  ^4-  =  1,  -^  +  '^  —  1 5  that  is,  the 
a-        ¥  a-        0- 


XyV  ?/,?/ 

equations  — 5-  + '  ' 
a- 


b' 


1    ^^  +  -M! 
'     a-  b- 


1  must  be  true.     Hence 


the  points  of  contact  lie  in  the  line 


+f=i, 


diich 


therefore  the  chord  of  contact.     Similarly,  it  is  found  that  the 

X-       "■' 
points  of  contact  from  (.r'l,  y/j)  to  the  hyperbola  -r,  - 


1  and 


yih 


to  the  parabola  y'  —  lpx  lie  in  the  lines  -—2^  —  72^  =  1  a,nd 
?///,  =  j>(;«  -f-  x^  respectively.  The  coordinates  of  the  points  of 
contact  of  tangents  through  (.»■„  v/j)  to  a  conic  section  are  found 


prxOPEUTIES   OF  THE  CONIC  SECTIONS  115 

by  making  the  equations  of  the  chord  of  contact  and  of  the 
conic  section  simultaneous  and  solving  for  x  and  //. 

A  theory  of  poles  and  polars  with  respect  to  any  conic  sec- 
tion might  be  constructed  entirely  analogous  to  the  theory  of 
poles  and  polars  with  respect  to  the  circle. 

The  equation  y  =  mx  +  n,  where  m  is  a  fixed  constant,  n  a 
parameter,  represents  a  system  of  parallel  straight  lines.  For 
any  value  of  n,  the  abscissas  of  the  points  of  intersection  of 
straight  line  and  ellipse  %,-\-%  =  l  are  found  by  solving  the 
equation  (b'  +  aha-)  x-  +  2  a-mnx  +  a"  (a-  -  b-)  =  0.  These  ab- 
scissas are  equal,  and  the  line  ?/  =  nix  +  n  becomes  a  tangent 

to   the   ellipse    -;,  +  -'^  =  l    when    u' =  b'- +  a-ni-.      Therefore 
u-      b- 

y  =  mx  ±  (b-  +  a-m-y  are  the  two  tangents  to  the  ellipse  whose 

slope  is  m.     In  like  manner  it  is  found  that  the  tangents  to 

the  hyperbola  whose  slope  is  m  are  y  =  nix  ±(a-iii'  —  b'-)- ;  the 

P 

tangent  to  tlu;  parabola  whose  slope  is  vi  is  y  —  rax  +  — — 

The  equations  of  the  normals  to  ellipse,  hyperbola,  and  parab- 
ola at  the  i)oint  (.»•„,  ?a,)  of  the  curves  are  y  —  7j,,  = -^(x  —  x^^, 
y  -?/„  =  -  "/"(.f  -  .f,),  //  -  //„  =  -  -''^(x  -  .Vu)  respectively. 

Problems.  —  1.  Find  the  eiiuatioiis  of  taiip:cnts  to  the  ellipse  whose 
axes  arc  S  and  0  at  the  points  wliose  distance  from  the  T-axis  is  1. 

2.  Find  the  eiiuatioiis  of  the  focal  tangents  of  ellipse,  hyperbola,  and 
parabola. 

3.  From  the  point  (fi,  8)  tangents  are  drawn  to  tlu'  ellipse  ^-|-^=1. 
F'ind  the  coordinates  of  the  points  of  contact  and  the  equations  of  the 
tangents. 

4.  At  what  point  of  the  parabola  ?/-  =  10x  is  the  slope  of  the  tan- 
gent 1  h  ? 

5.  On  an  elliptical  track  whose  major  axis  is  due  east  and  west  and  1 
mile  long,  minur  axis   !   mile  lonsr,  in  what  direction  is  a  man  traveling 


116  ANALYTIC  GEOMETRY 

when  walking  from  west  to  east  and  ]  mile  west  of  the  north  and  south 
line  ? 

6.  Write  the  equations  of  tangents  to  ^  +  ^  =  1  making  an  angle  45° 
with  the  X-axis. 

7.  Write  the  equations  of  the  tangents  to  —  —  ^  =  1  perpendicular  to 
2x-32/  =  4.  ^       ^ 

8.  Write  the  equation  of  the  tangent  to  y-  =  8x  parallel  to  ^  +  -^  =  ^• 

9.  Find  the  slopes  of  the  tangents  to  — +  ^- =  1  through  the  point 

9       4 

(4,5).     ?/  =  mx  +  (4  +  9m2)2  is  tangent  to— +  ^=  1.    Since  (4,  5)  is  m 

i  9       4 

the  tangent,  5  =  4  wi  +  (4  +  9  m^)  2.     Solve  for  m. 

10.  Find  the  slopes  of  tangents  to  ^  -  ^  =  1  through  (2,  3). 

11.  Find  the  slopes  of  tangents  to  y'  =  Gx  through  (-5,  4). 

12.  Find  the  points  of  contact  of  tangents  to  y"  =  {Jx  through  (  -5,  4). 

13.  Find  the  intercepts  of  normals  to  ellipse,  hyperbola,  and  parabola 
on  X-axis. 

14.  Find  distances  from  focus  to  point  of  intersection  of  normal  with 
axis  for  each  of  the  conic  sections. 

15.  Prove  that  tangents  to  ellipse,  hyperbola,  or  parabola  at  the  ex- 
tremities of  chords  through  a  fixed  point  intersect  on  a  fixed  straight  line. 

16.  Prove  that  the  chords  of  contact  of  tangents  to  a  conic  section 
from  points  in  a  straight  line  pass  through  a  common  point. 

17.  Show  that  the  tangent  to  the  ellipse  at  any  point  bisects  the  angle 
made  by  one  focal  radius  to  tlie  point  with  the  prolongation  of  the  other 
focal  radius  to  the  point. 


rUOVERTIKS   OF  THE   CONIC   SECTIONS 


117 


The  ratio  of  the  focul  nulii  is  "^^  =  - — ^-     Since 
PF'      a  +  exo 


AF=  AF'  =  ae  and  AT- 


Xo    F'T~  a 


(«  -  ex^) 


—  («  +  ea;o) 


Hence  EJL  =  I1L^  and  Pr  bisects  FPS. 
F'T     PF' 


18.  In  the  hyperbohx  the  tangent  at  any  point  bisects  the  angle  in- 
cluded by  the  focal  radii  to  the  point. 

19.  In  the  parabola  the  tangent  at  any  point  bisects  the  angle  included 
by  the  focal  radius  to  and  the  diameter  through  the  point.* 


\D' 


Fig.  104. 


On  problems  17,  18,  19  is  based  a  simple  method  of  drawing  tangents 
to  the  conic  sections  through  a  given  point.  With  the  given  point  as 
center  and  radius  equal  to  distance  from  given  point  to  one  focus  strike 

*  Since  it  is  true  of  rays  of  light,  heat,  and  sound  that  the  reflected  ray 
and  the  incident  ray  lie  on  different  sides  of  the  normal  and  make  equal 
angles  with  the  normal,  it  follows  that  rays  emitted  from  one  focus  of  an 
elliptic  reflector  are  concentrated  at  the  other  focus ;  that  rays  emitted 
from  one  focus  of  an  hyperbola  reflector  proceed  after  reflection  as  if 
emitted  from  the  other  focus  ;  that  rays  emitted  from  the  focus  of  a 
parabolic  reflector  after  reflection  proceed  in  parallel  lines. 

It  is  this  property  of  conic  sections  that  suggested  the  term  focus  or 
"  burning  point." 


118 


A NA L  YTIC  GEOMETR  Y 


off  an  arc.  In  the  parabola  the  parallels  to  the  axis  through  the  inter- 
sections of  this  circle  with  the  directrix  determine  the  points  of  tan- 
gency.  For  TF  =  TD,  hence  the  triangles  TPF,  TPD  are  equal  and  PT 
is  tangent  to  the  parabola.  In  ellipse  and  hyperbola  strike  off  another 
arc  with  the  second  focus  as  center  and  radius  equal  to  transverse  axis. 


Lines  joining  the  second  focus  with  the  points  of  intersection  of  the  two 
arcs  determine  the  points  of  tangency.  In  the  ellipse  T'F'  +  T' F  =  2  a, 
and  by  construction  T'F'  +  TD'  =  2  a,  hence  T'F  -  T'D'.  The  trian- 
gles T'PF,  T'PD'  are  equal,  and  PT'  is  tangent  to  the  ellipse.  In 
the  hyperbola  TF'  -  TF  =  2 a,  TF'  -  TD  =  2a;  hence  TD  =  TF,  the 
triangles  TPD,  TPF  are  equal,  and  PT  is  tangent  to  the  hyperbola. 

20.  Show  that  the  locus  of  the  foot  of  the  perpendicular  from  the  focus 

of  the  ellipse \--l-=zl  to  the  tangent  is  the  circle  described  on  the 

a-     b'^ 
major  axis  as  diameter. 

The  equation  of  the  perpendicular  from  the  focus  {ae,  0)  to  the  tan- 
gents y  =  mx  ±  (b-  +  n-m^)  •2  is  my  +  x  =  ae.  Make  these  equations 
simultaneous  and  eliminate  m  by  squaring  both  equations  and  adding. 
There  results  x'^  +  y^  =  a'^. 

21.  Show  that  the  locus  of  the  foot  of  the  perpendicular  from  the  focus 

of  the  hyperbola  ^  -  ^-  =  1  to  the  tangent  is  the  circle  described  on  the 

a^      b'^ 
transverse  axis  as  diameter. 

22.  Show  that  the  locus  of  the  foot  of  the  perpendicular  from  the  focus 
of  the  parabola  y^  =  2pxto  the  tangent  is  the  F-axis. 


PUOI-KliTIKS    or   THE   CONIC   SECTIONS 


119 


Problems  20,  21,  22  may  be  used  to  construct  the  cuuic  sections  as 
envelopes  when  the  focus  and  the  vertices  are  known. 

23.  Prove  that  for  ellii)se  and  hyperbola  the  product  of  the  perpen- 
diculars from  foci  to  tangent  is  constant  and  eipial  to  h'-. 

24.  Prove  that  in  the  parabola  the  locus  of  the  point  of  intersection  of 
a  line  through  the  vertex  perpendicular  to  a  tangent  with  the  ordinate 
through  the  point  of  tangency  is  a  semi-cubic  parabola. 


AkT.    62. (JONJUGATK    DiAMETERS 


Let  {xo,  ?/„)  be  the  point  of  intersection  of   the  diameter 

w  =  tan^  •  X  with  the  ellipse  — \-^^=l,  and  call  the  angle 
a'     Ir 

made  by  the  tangent  to  the  ellipse  at  (.r,,,  ij^^)  with  the  X-axis  $\ 

Then 

Y 


tan  d  =  •^, 

tan^'  =  -^ 

tan  6  tan  6'  =  - 

Now  let  (a'l,  ?/,)  be  the  point 
of  intersection  of 

Fig.  107. 

y  =  tan  9'  -  x 

with  the  ellipse,  and  call  the  angle  made  by  the  tangent  to  the 
ellipse  at  (a;,,  ?/i)  with  the  X-axis  0.     Then 


tan 


?/i 


tan  e=^-  ^,  tan 


tan  6'  =  - 


Ir 


Hence  the   condition   tan  6  tan  &  = 


causes   each    of    the 


diameters  of  the  ellipse  y  =  tan  0  •  x,  y  =  tan  0'  ■  x  to  be  par- 
allel to  the  tangent  at  the  extremity  of  the  other.  Such 
diameters  are  called  conjugate  diameters  of  the  ellipse. 


120 


ANALYTIC   GEOMETRY 


The  equation  of  tlie  ellipse  —-{---^^^l  referred  to  a  pair  of 
(r      b'- 
conjugate  diameters  and  in  terms  of  the  semi-conjugate  diam- 
eters a'  and  b'is—  +  ^=l.     (See  Art.  35,  Prob.  39.)     This 

equation  shows  that  each  of  a  pair  of  conjugate  diameters 
bisects  all  chords  parallel  to  the  other.  The  axes  of  the 
ellipse  are  a  pair  of  perpendicular  conjugate  diameters. 

Let  (xu,  2/o)  be  the  point  of  intersection  oi  y  —  tan  6  •  x  with 
the  hyperbola  —  —  ^  =  1,  and  call  the  angle  made  by  the  tan- 
gent  to  the  hyperbola  at  {xq,  y^)  with  the  X-axis  6'.      Then 
.Vo    4-„„  flf  _  ^^•^•o    tan  6*  tan  6>' =  — .      Since    y  =  ~x    and 

b 

y  — X     are 

a 


tan  0^'^,   tan  0' 
x„ 


b%, 


tl 


le     common 


asymptotes  of  the  pair  of  con- 
jugate hyperbolas 


and  —  — - 


cr      b'' 
l,it  is  evident 


that  the  condition 

tan  d  tan  $' =  — 

causes  y  =  tan  0'  •  x  to  inter- 
sect--^=-1  if 


tan  I 


intersects  —-■£=!.     Now  suppose  (x^,  y^  to  be  the  point  of 

a-    y- 

intersection  of  the  line  y  =  tan  6'  •  x  with  the  conjugate  hyper- 


bola 


1,  and  call  the  angle  made  by  the  tangent  to 


this  hyperbola  at  (.Ti,  ?/i)  with  the  X-axis  6.     Then  tan  & 


tan 


b-x, 


tan  e  tan  6'  = 


Diameters    of    the    hyperbola 
b^ 


satisfying  the  condition  tan  6  tan  0'  =-7,  are  called  conjugate 
diameters  of  the  hyi^erbola. 


PliOPEliTIEtS   OF  THE  CONIC  SECTIONS 


121 


The  ecjuation  of  the  hyperbola  '- 
of  conjugate   diameters 
diameters  a'  and  b',  is  -^.  —  r^=l.      (See  Art 


cr      Ir 
and   in   terms   of   tl 


referred  to  a  pair 

le    semi-conjngate 

Prob.  38.) 


This  equation  shows  that  chords  of  an  hyperbola  parallel  to  any 
diameter  are  bisected  by  the  conjugate  diameter.  The  axes  of 
the  hyperbola  are  perpendicular  conjugate  diameters. 

The  equation  of  the  parabola  referred  to  a  diameter,  and  a 
tangent  at  the  extremity  of 
the  diameter,  is  y^  —  2piX. 
(See  Art.  35,  Prob.  40.)  This 
equation  shows  that  any  diam- 
eter of  the  parabola  bisects 
all  chords  parallel  to  the  tan- 
gent at  the  extremity  of  the 
diameter.  The  axis  of  the 
parabola  is  that  diameter 
which  bisects  the  system  of 
parallel  chords  at  right  angles. 

It  is  now  possible  to  deter- 
mine geometrically  the  axes,  focus,  and  directrix  of  a  conic 
section  when  the  curve  only  is  given.  In  the  case  of  the 
ellipse  draw  any  pair  of  parallel  chords.  Their  bisector  is  a 
diameter  of  the  ellipse. 
With  the  center  of  the 
ellipse  as  center  strike 
off  a  circle  intersecting 
the  ellipse  in  four  points. 
The  bisectors  of  the  two 
pairs  of  parallel  chords 
joining  the  points  of  in- 
tersection   are    the    axes 

of  the  ellipse.  An  arc  struck  off  with  extremity  of  minor 
axis   as   center,  and   radius   equal  to  semi-major   axis,  inter- 


122 


ANALYTIC   GEOMETRY 


sects  the  major  axis  in  the  foci.     The  directrix  is  perpendicular 
to  the  line  of  foci  where  the  focal  tangents  cross  this  line. 

In  the  case  of  the  hyperbola  the  directions  of  the  axes  are 
found  as  for  the  ellipse.  The  focus  is  determined  by  drawing 
a  perpendicular  to  any  tangent  at  the  point  of  intersection  of 
this  tangent  with  the  circumference  on  the  transverse  axis. 
Drawing  the  focal  tangents  determines  the  directrix.  The 
conjugate  axis  is  limited  by  the  arc  struck  off  with  vertex  as 
center  and  radius  equal  to  distance  from  focus  to  center. 

In  the  case  of  the  parabola, 
after  determining  a  diameter 
by  bisecting  any  pair  of  paral- 
lel chords,  and  the  axis  by 
bisecting  a  pair  of  chords  per- 
pendicular to  the  diameter, 
the  focus  is  determined  by 
the  property  that  the  tangent 
bisects  the  angle  included  by 
diameter  and  focal  radius  to 
point  of  tangency. 


Art.  63.  —  Supplementary  Chords 

Chords  from  any  point  of  ellipse  or  hyperbola-to  the  extrem- 
ities of  the  transverse  axis  are   called   supplementary.     Let 

(x',  y')  be  any  point  of  the 


The 


ellipse   ^„  +  j-^ 

equations  of  lines  through 

(a;',  ?/'),  (a,  0)  and  (.-»',  ?/'), 
( —  a,  0)  are 


Fig.  112. 


•/      {X  -  a), 


x'  —  a 

,    y' 


PliOPKRTlES   OF   THE  CONIC  SECTIONS 


123 


Calling  the  angles  made  l)y  tlie  supplementary  eliords  with 


the  X-axis  </>  and  <^',  tan  <^  tan  4>'  — 


V 


From  the  equa- 


x'-).    Hence  tan  </>  tan  (/>'  = ;. 

d- 

—  =  1 ,  tan  <f)  tan  </>  =  — . 

a-      Ir  a' 

V  are  a  pair  of  conjugate  diam- 

1  when  tan  0  ■  tan  $'  =  —  —.    Hence 
d- 


tion  of  the  ellipse,  //'-  =  —(a' 

In  like  manner  for  the  hyperbola 
y  =  tan  6  •  x  and  ?/  =  tan  6 

eters  of  the  ellipse  '—4--^-- 
d-  h- 
tan  6  •  tan  6'  =  tan  4>  •  tan  </>', 
from  which  it  follows 
that  if  one  of  a  pair  of 
supplementary  chords  is 
parallel  to  a  diameter  the 
other  chord  is  parallel  to 
the  conjugate  diameter. 
This  proposition  is  dem- 
onstrated for  the  hyper-  ' 
bola  in  the  same  manner. 

On  this  proposition  are  based  simple  methods  of  drawing 
tangents  to  ellipse  or  hyperbola,  either  through  a  point  of  the 
curve  or  parallel  to  a  given  line.  To  draw  a  tangent  to  the 
ellipse  at  any  point  P,  dra^v  a 
diameter  through  P,  a  sup- 
plementary chord  parallel  to 
this  diameter,  and  the  line 
through  /'parallel  to  the  other 
supplementary  chord  is  the 
tangent. 

To  draw  a  tangent  to  the 
hyperbola  parallel  to  a  given 
straight  line,  draw  one  sup- 
plementary chord  parallel  to  the  given  line,  and  the  diameter 
parallel  to  the  other  supplementary  chord  determines  the  points 
of  tangency. 


124  ANALYTIC   GEOMETEY 

To  draw  a  pair  of  conjugate  diameters. of  an  ellipse,  includ- 
ing a  given  angle,  construct  on 
^^^^^^         /^^^  \  ^^^  major  axis  of  the  ellipse  a 

/^^     ^--V  nh  circular  segment  containing  the 

— x ^""^^^  /     "^  ~^  ^  -J ^        given  angle.     From  the  point  of 

/ V  /'  - -^^^^T"^-----/!        intersection  of  the  arc  of  the  seg- 

1  ^v^y^  1^      ^^  ]\f)     ment  and  the  ellipse  draw  a  pair 

I     "^  /  \      of   supplementary   chords.     The 

PiQ  115  diameters  parallel  to  these  chords 

are  the  required  diameters. 

Art.  64.  —  Parameters 
Since  —  +  ^  =  1  is  the  equation  of  an  ellipse  referred  to  any 

pair  of  conjugate  diameters,  it  is  readily  shown  that  the 
squares  of  ordinates  to  any  diameter  of  the  ellipse  are  in  the 
ratio  of  the  rectangles  of  the  segments  into  which  these  ordi- 
nates divide  the  diameter.  The  same  proposition  is  true  of 
the  hyperbola. 

Taking  the  pair  of  perpendicular  conjugate  diameters  of  tlie 
ellipse  as  reference  axes  and  the  points  (cte,  'p),  (0,  IS),  the 
proposition   leads   to  the    proportion  ^  =  — ^^ — v^^^y    whence 

-^  —  —  -  that  is,  the  i:)arameter  to  the  transverse  axis  of  the 
2&      2a'  '  _ 

ellipse  is  a  fourth  proportional  to  the  transverse  and  conjugate 
axes.  Generalizing  this  result,  the  parameter  to  any  diaineter 
of  ellipse  or  hyperbola  is  the  fourth  proportional  to  that  diame- 
ter and  its  conjxigate. 

In  the  common  equation  of  the  parabola,  y''-  =  2i')X,  the 
parameter  2p  is  the  fourth  proportional  to  any  abscissa  and 
its  corresponding  ordinate.  Generalizing  this  definition,  the 
parameter  to  any  diameter  of  the  parabola  is  the  fourth  propor- 
tional to  any  abscissa  and  its  corresponding  ordinate  with 
respect  to  this  diameter. 


iniOPEIiTIES    OF  THE   CONIC  SECTIONS 


125 


^Vhen  (in,  v)  on  the  parabola  y"  =  2px-  is  taken  as  origin,  the 
diameter  througli  {m,  n)  as  X-axis,  the  tangent  at  (?//.,  n)  as 
l''-axis,  the  e(i[uation  of  the  pa- 
rabola takes  the  form  Y 


yf  =  -J2h^i- 


(See  Art.  35,  I'rob.  40.) 


^'      sure 


Hence  22h  =  4(?ji  +  J- jj);  that 
is,  the  parameter  to  any  di- 
ameter of  a  parabola  is  four 

times  the  focal  radius  of  the  vertex  of  that  diameter.     Calling 
the   focal   radius  /,    the   equation   of   the    parabola    becomes 


r_ 


1,   find  the   equation   of    the 


Problems.  —  1.    In  the  ellipse 
rlianieter  conjugate  to  y  =  x. 

2.  Find  the  angle  between  the  supplementary  chords  of  the  ellipse 

'■!.'  ^111=1  at  the  extremity  of  the  minor  axis. 
n-     //- 

3.  Find  the  point  of  the  ellipse  ^  +  ^'-1  at  which  supplementary 
chords  include  an  angle  of  45°. 

4.  Show  that  the  maximum  angle  between  a  pair  of  supplementary 
x'  ,  if      ,   ..„  ..„_,    2  ah 

h 


chords  of  the  ellipse 


1  IS  tan-' 


«•'     o~  ¥  —  a- 

5.  Show  that  a  pair  of  conjugate  diameters  of  an  hyperbola  cannot 
include  an  angle  greater  than  90°. 

6.  Construct  the  ellipse  whose  equation  referred  to  a  pair  of  conjugate 


-f^ 


Find  focus  and  dircc- 


diameters  including  an  angle  of  45° 
trix  of  this  ellipse. 

7.  Find  the  equation  of  the  hyperbola  whose  axes  arc  8  and  6  re- 
ferred to  a  pair  of  conjugate  diameters,  of  which  one  makes  an  angle  of 
45°  with  the  axis  of  the  hyperbola.  Find  lengths  of  the  semi-conjugate 
diameters. 


126 


ANALYTIC   GEOMETRY 


8.    Find  equation  of  parabola  whose  parameter  is 
ter  through  (8,  8)  and  tangent  at  this  point. 


Find  the  locus  of  the  centers  of  chords  of 


2x 


referred  to  diame- 


^  =  1  parallel  to 
4 


is  y^=- 


a  parabola   referred  to 


10.  The  equation  of  a  pai-abola  referred  to  oblique  axes  including  an 
angle  of  60°  isy-  =  lOx.  Sketch  the  parabola  and  construct  its  focus  and 
directrix. 

11.  A  body  is  projected  from  A  in  the  direction  AY  with  initial 
velocity  of  v  feet  per  second.  Gravity  is  the  only  disturbing  force.  Find 
the  path  of  the  body  and  its  velocity  at  any  instant. 

Taking  the  line  of  projection  as  F-axis  and  the  vertical  through  A  as 

X-axis,   the  coordinates  of  the  body   t  seconds   after    projection    are 

x  =  I  gfi,    y  =  vt;    the  equation  of   the 

path  of  the  body,  found  by  eliminating  t, 

(J 

tangent  and  diameter  through  point  of 
tangency.  Comparing  this  e<iuation  with 
2/2  =  4 /x,  the  equation  of  parabola  re- 
ferred to  tangent  and  diameter,  v'^=2  (jf ; 
that  is,  the  initial  velocity  is  the  velocity 
acquired  by  a  body  falling  freely  from 
the  directrix  of  the  parabola  to  the  start- 
ing point. 

If  the  body  is  projected  from  any 
point  of  the  parabola  along  the  tangent 
to  the  parabola  at  that  point,  and  with  a  velocity  equal  to  the  velocity  of 
the  body  projected  from  A  wlien  it  reaches  that  point,  the  path  of  the 
body  is  the  path  of  the  body  projected  from  A.  Hence  it  follows  that 
the  velocity  of  the  body  at  any  point  of  the  parabola  is  the  velocity 
acquired  by  a  body  freely  falling  from  the  directrix  of  the  parabola  to 
that  point. 


Art.  65.  —  The  Elliptic  Compass 


Let  i^  -f  -^  =  1  aud  3l  +  -^^  =  1  1)0,  two  ellipses  ccmstructed 
on  the  same  major  diameter.-  Let  ?/,  and  ?/.  be  ordinates  cor- 
respondintr  to  the  same  abscissa,  then  ■—  =  ^^;  that  is,  if  ellipses 


riiOPEIiTIES   OF  THE   CONIC  SECTIONS 


127 


are  constructed  on  the  same  major  diameter,  corresponding 
ordinates  are  to  each  other  as  the  minor  diameters.  The  circle 
described  on  the  major  diameter  of  the  ellipse  is  a  variety  of 
the  ellipse,  hence  the  ordinate 
of  an  ellipse  is  to  the  corre- 
sponding ordinate  of  the  cir- 
cumscribed circle  as  the  minor 
diameter  of  the  ellipse  is  to 
the  major  diameter. 

On  this  principle  is  based 
a  convenient  instrument  for 
drawing  an  ellipse  whose  axes 
are  given.  On  a  rigid  bar 
take  PH=a,  PK^h.  Fix 
pins  at  H  and  K  which  slide  in  grooves  in  the  rulers  X  and  "J 

perpendicular  to  each  other 
Fo 


lY            P' 

. 

^.''- 

' — 

^< 

^  ' 

1       \ 

1         \ 

/ 

/ 

/      / 

1        \^ 

x\ 

^/  ^ 

D        i^ 

1 

■"^^^ 

/ 

H        ^y 

^~~^ 

Y' 

PI)     PII 

compass. 


P  traces  the  ellipse  ^  -f  ^  =  1 . 
This   instrument   is  called  the  elliptic 


Art.  66.  —  Area  of  the  Ellipse 

Erect  any  number  of  perpendiculars  to  the  major  diameter 
of  the  ellipse,  and  beginniug  at  the  right  draw  through  the 
points  of  intersection  of  these 
perpendiculars  with  the  ellipse 
and  the  circumscribed  circle 
parallels  to  the  minor  diam- 
eter. There  is  thus  inscribed 
in  the  ellipse  and  in  the  circle 
a  series  of  rectangles.  The 
corresponding  rectangles  in 
ellipse  and  circle  have  the 
same  base,  and  their  altitudes 
are  in   the  ratio  of    h  to  a. 


128 


A NAL  YTIC   GEOMETR  Y 


Hence  the  sum  of  the  areas  of  the  rectangles  inscribed  in 
the  ellipse  bears  to  the  sum  of  the  rectangles  inscribed  in  the 
circle  the  ratio  of  h  to  a.  By  indefinitely  increasing  the 
number  of  rectangles,  the  sum  of  the  areas  of  the  rectangles 
inscribed  in  the  ellipse  approaches  the  area  of  the  ellipse  as  its 
limit,  and  at  the  same  time  the  sum  of  the  areas  of  the  rec- 
tangles inscribed  in  the  circle  approaches  the  area  of  the  circle 

as  its  limit.     At  the  limit  therefore  ^^-^ r-^^  =  -,  hence 

,  area  of  circle       a 

area  of  ellipse  =  -  •  7ra^  =  irah. 
a 


Art.  67.  —  Eccentric  Angle  op  Ellipse 


At  any  point  {x,  y)  of  the  ellipse  ^,  +  4,  ■ 


1  produce  the 


ordinate  to  the  transverse  axis  to  meet  the  circumscribed  circle 
and  draw  the  radius  of  this  circle  to  the  point  of  meeting. 

The  angle  <^  made  by  this 
radius  with  the  transverse 
axis  of  the  ellipse  is  called 
the  eccentric  angle  of  the 
point  (.r,  ?/).  Erom  the  figure 
x—.a  •  cos  <)!>, 

y  =  -.    1\D=^  -  ■  asinc^ 
((  a 

=  h  •  sin  <^. 
The  coordinates  of  any  point 
{x,  ?/)  of  the  ellipse  are  thus 
expressed  in  terms  of  the 
single  variable  ^. 
Let  AP^  and  AP^  be  a  pair  of  conjugate  diameters  of  the 
1,  6  and  d'  the  angles  these  diameters  make 

^'.      Let 


ellipse  I +  |: 

Avith  the  axis  of  the  ellipse.       Then  tan  6  tan  6' 


I'llOPKUTIES    OF  THE  CONIC   SECTIONS 


129 


(.I'l,  ?/i)  be  the  coordinates,  ^i  the  eccentric  angle  of  J\;  {.i:,,y.,) 
tlie  coordinates,  <f>.2  the  eccentric  angle  of  F2.     Then 


tan  e 


_?/,  _  h  sin 


•i'l 

a  cos  ^1 

tai 

X., 

_  6  sin  <^o 

<<-  cos  </)j 

tan  e  tan  0' 

h-  sin 

<^i  sin  <)!)2 

a-  cos 

</>!  cos  ^2 

</>itan^2 

-      ^'l 

(r 

Hence  tan  c^,  tan  <^,  =  -  1  and  c/,,  and  <^.  differ  l.y  00°;  that  is, 
the  eccentric  angles  of  the  extremities  of  a  pair  of  conjngate 
diameters  of  the  ellipse  differ  by  90°. 

Call  the  lengths  of  the  semi-conjugate  diameters  a,  and  />,. 

Then  a{  =  .c,-  +  v/f  =  or  cos''  <^i  +  Ir  sin-  ^i, 

h{  =  a-  cos-  (f)-.  +  b''  sin^  (f>2  =  «^  sin^  ^1  +  h-  cos^  ^1, 

since  cj^.  =  90°  +  <^i.     T.y  addition  ac  +  b^'  =  (r  +  b-;  that  is, 
the  sum  of  the  squares  of  any  pair  of  conjugate  diameters  of 
the  ellipse  equals  the  sum  of  the  squares  of  the  axes. 
The  conjugate  diameters  are  of  equal  length  when 

a-  cos-  <i>  -\-b-  sin-  cji  =  a-  sin'  <f)  +  h-  cos^  cf> ; 

that  is,  when 

tan-  <^  =  1,  tan  </>  =  ±  1,  <^  =  45°  or  13")°. 

The  cipiationsof  the  cc^ual  conjugate  diameters  are  y  =  ±~x, 

and  their  length   Vw(u'  +  b'-). 

The   area   of    the    parallelogram    circumscribed   about    the 

K 


130 


.1 NAL  VTIC   G EOMETll  Y 


ellipse  with  its  sides  parallel  to  a  pair  of  conjugate  diameters 

is  4  6'  •  AN.  The  equation 
of  the  tangent  to  the  el- 
lipse at  (x',  y')  is 

xx'      ?///'  _  ^ 

The  point  (x',  y')  is  the 
same  as  (a  cos  ^j,  h  sin  <^i), 
and  the  tangent  may  be 
written 


X  cos  <^i      y  sin  c^j 


1. 


The   length   of    the   perpendicular   from    the    origin   to   this 
tangent  is        AJS  — 


/cos-  </>!      sin-</)|\  -       ^1 
1^     a?     ^      U^    ) 


Hence  4  6'  •  AN=  4a6;  that  is,  the  area  of  the  circumscribed 
parallelogram  equals  the  area  of  the  rectangle  on  the  axes. 


Art.  68.  —  Eccentric  Angle  of  the  Hyperbola 

On  the  transverse  axis  of  the  hyperbola  describe  a  circle. 
Through  the  foot  of  the  ordinate  of  any  point  (x,  y)  of  the 
hyperbola  draw  a  tangent  to  this 
circle ;  the  angle  made  by  the 
radius  to  the  point  of  tangency 
and  the  axis  of  the  hyperbola  is 
called  the  eccentric  angle  of  the 
point  {x,  y).  From  the  hgure 
x  =  a  •  sec  ^  and,  since 

7.2 

y-  —    _^  (a-  —  X-),  y  =  h  •  tan  4>. 


Let  Al\  and  AP-.  be  a  pair  of  conjugate  diameters  of  the 


I'llOl'EliTlES    OF   THE  CONIC  SECTIONS 


131 


hyperlxtla 


1;   (.i\,i/i)  the  coordinates,  c/),  the  eeeeutric 


angle  of  the  point  I\  ;  6  and 
0'  the  angles  included  by  the 
conjugate  diameters  and  the 
axes  of  the  hyperbola.     Then 

b  tan  <^, 


tan 


.Tj      (( sec  </>! 


Since    tan  6  tan  $'  =     , 
tane'= r 


a  sm  (pi 

Hence   the   equations   of   the 
conjugate  diameters  are  y  = ^ 

point  of  intersection    of    ?/  = 


x,y  = 


a  sm 

X- 


dtl 


Po,  th 
-1,  i 


a  sin  ^1  cr     0- 

(((tan^i,  6secc^i).  Therefore  APi  —  Ui' =  a- sec"^  cl>  +  b- tmr  4>, 
AFi  =  bi^  =  a^  tan^  <^i  +  b^  sec-  <^i.  By  subtraction  a^-  —  bc 
=  a^  —  b-;  that  is,  the  difference  between  the  squares  of  any 
pair  of  conjugate  diameters  of  the  hyperbola  equals  the  differ- 
ence of  the  squares  of  the  axes. 

The  area  of  the  parallelogram  whose  sides  are  tangents  to  a 
pair  of  conjugate  hyperbolas  at  the  extremities  of  a  pair  of  con- 
jugate diameters  is  Aby  AN.     The  equation  of  the  tangent  to 

x^     y-      i      ,   ,  ,     7  i.       ,  \  •     sec  <f>.         tan  Aj  ,      ^      mv,r> 

—  —  ^=1  at  (a  sec  cb^,  b  tan  <ii)  is  ^.r --^y  =  1.     ihe 

a^      b'^  a  b 

perpendicular  from  tlie  origin  to  this  tangent  is 


AN: 


+ 


tan-  </>, 


•a-  b'- 

Hence  the  area  of  the  parallelogram  equals  iab;  that  is,  the 
area  of  the  rectangle  on  the  axes. 


sec<^..^. 
a 

tan  (^, 
6        '' 

-  sec  </,, 

..+-^.. 

a 

tanc^,_^ 
a 

sec  <^i       _ 
b       -^ 

—  tan  <^i 
a 

■x  +  '-^^^-^.y 

132  ANALYTIC  GEOMETRY 

The  equations  of  the  sides  of  the  parallelogram  are 

1,  (1) 

1,  (2) 

1,  (3) 

-  1.  (4) 

Making  these  equations  simultaneous  and  combining  by  addi- 
tion or  subtraction,  it  is  found  that  the  vertices  of  the  parallelo- 
gram lie  in  the  asymptotes   y  =  ±  -x. 

Problems.  —  1.    Find  the  area  of  the  ellipse  whose  axes  are  8  and  6. 

2.  What  are  the  eccentric  angles  of  the  vertices  of  the  ellipse  ?  of  the 
ends  of  the  focal  ordinate  to  the  transverse  axis  ? 

3.  The  extremity  of  a  diameter  of  the  ellipse  — -|-f-=  1   is  (xi,  yi), 

a-'     0^ 

the  extremity  of  the  conjugate  diameter  (x2, 2/2)-  Find  X2  and  2/2  in  terms 
of  Xi  and  yi. 

4.  Solve  the  same  problem  for  the  hyperbola. 

5.  In  the  hyperbola  whose  axes  are  10  and  0  the  length  of  a  diameter  is 
15.     Find  the  length  of  the  conjugate  diameter. 

6.  Find  the  lengths  of  the  equal  conjugate  diameters  of  the  ellipse 
whose  axes  are  12  and  8.  Also  the  equation  of  this  ellipse  referred  to  its 
equal  conjugate  diameters. 


CHAPTER   X 

SECOND  DEGKEE  EQUATION 

AuT.  69.  —  Locus  OF  Second  Deouee  Equation 

Write  the  general  second  degree  equation  in  two  variables 
in  the  form 

(ur  +  2  bx!f  +  qf  +  2  dx  +  2  ey  +  /  =  0.  (1) 

The  problem  is  to  determine  the  geometric  iigure  represented 
by  this  equation  when  interpreted  with  respect  to  the  rectangu- 
lar axes  X,  Y.  The  equation  of  this  geometric  figure  when 
referred  to  axes  Xj,  \\,  parallel  to  X,  Y,  with  origin  at  (a-o,  y^, 
becomes 
aa;/  +  2  bx,y,  +  cy,-  +  2  {ax,  +  hy^  +  d)x,  +  2  {bx,  +  ry,  +  e)y, 
+  ax,'  +  2  bx„yo  +  cy,'  +  2  dx,  +  2  cy,  +f=  0.  (2) 

The  geometric  figure  is  symmetrical  with  respect  to  the  new 
origin  (a^o,  y^  if  the  coefficients  of  the  terms  in  the  first  powers 
of  the  variables  in  equation  (2)  are  zero.  The  coordinates  of 
the  center  of  symmetry  of  the  figure  are  therefore  determined 
by  the  equations  ax,  +  by,  +  fZ  =  0,  bx,  +  c?/o  +  e  =  0.  Whence 
^  eb  -  cd^  ^  db  -  ae  rj.^^  center  is  a  determinate  finite 
ac  —  b-  ac  —  ¥ 

point  only  when  ac  —  b'  ^  0. 

Suppose  ac  —  b^  =^  0.     The  absolute  term  of  ecpiation  (2)  be- 
comes 

ax,-  +  2  6a-ov/o  +  c?/o'  +  2  dx,  -\-2eyo+f 

=  Xo(axo  +  by,  +  (Z)  +  2/o  (c?/o  +  ^.I'o  +  <')+  dx,  +  ey,  +  f 

,      ,  ,    ^     acf+2bde-ae--cd'-fb- 

=  dx,  +  c!/o  +  /  =  -"^^^^ TV, 

ac  —  b' 

133 


134  ANALYTIC   GEOMETRY 

Writing  the  last  expression  ,  equation  (2)  becomes 

ac  —  b- 

axf  +  2  bx,y,  +  c?/f  +  —^^  =  0,  (3) 

ac  —  c>- 

or  axj-  +  2  6a;i?/i  +  ci/f  +  A;  =  0,  (4) 

where  k  —  dx,^  +  e^y,,  + ./". 

If  A  =  0,  equation  (3)  becomes 

ax{'  +  2  bx,ii,  +  ci/f  =  0,  (5) 

which  determines  two  values  real  or  imaginary  for  •— ;  that  is, 

the  equation  resolves  into  two  linear  equations,  and  hence 
represents  two  straight  lines.  An  e(|uation  which  resolves  into 
lower  degree  equations  is  called  reducible,  and  the  function  of 
the  coefficients,  A,  whose  vanishing  makes  this  resolution  pos- 
sible, is  called  the  discriminant  of  the  equation. 

Turn  the  axes  Xj,  Yi  about  the  origin  (xq,  ?/(,)  through  an 
angle  6.     Equation  (4)  becomes 

(a  cos^  e  +  c  sin-  ^  +  2  &  sin  ^  cos  6)x.f 

+  (a  sin-  6  -j-c  cos^  6  —  2b  sin  6  cos  6)yi 
+  2  { (c  -  a)  sin  ^  cos  ^  +  5(cos-  6  -  sin-  0)  I  x.fli.  +  fc  =  0. 
Determine    6    by  equating    to    zero    the    coefficient    of    x^^^ 

whence     tan  2  (9  =    '^  ^  .      Writing    the     res.ulting     equation 

a  —  c 
3Ixi  +  Ny^^  +  k  =  Q,  it  follows  that 

M+  N=  a  +  c,    il/-  JV=(a  -  c)cos(2 ^)+  2 6  sin(2^). 
From        tan  (2^)=-^,    sin  (2^)  = — -, 


cos  (2  6)  = '^ — ^ -• 

\^b'+{a-c)X' 

Therefore,  M-\-N=^a  +  c,  3/- iV^=  ^6' +(«  -  c)-Ss  and 
MN=  ac  —  b-.  Now  the  equation  3fx.f  +  Ny-r  +  A'  =  0  repre- 
sents an  ellipse  referred  to  its  axes  when  M  and  N  have  like 
signs,  an  hyperbola  referred  to  its  axes  when  31  and  N  have 


SECOND   DEGREE  EQUATION  135 

unlike  signs.     Hence  the  second  degree  equation  represents  an 
ellipse  when  ac  —  b'^>  0,  an  hyperbola  when  ac  —  b^  <  0. 

tan  (2^)=    ^       determines  two  values  for  2  $,  and  the  radi- 
a  —  c 

cal  \-ih-  +(a  —  c)-J  ^  has  the  double  sign.  To  resolve  the  ambi- 
guity take  2  6  less  than  180°,  which  makes  sin  (2  6)  positive,  and 
requires  that  the  sign  of  the  radical  be  the  same  as  the  sign  of  b. 
When  a  —  c  and  the  radical  have  the  same  sign,  cos  (2  9)  is  posi- 
tive and  2  6  is  less  than  90° ;  when  a  —  c  and  the  radical  have 
different  signs,  cos  (2^)  is  negative  and  2  6  is  greater  than  90°. 
The  ambiguity  may  be  resolved  and  the  squares  of  the  semi- 
axes  calculated  in  this  manner.     The  equation  tan  (2  6*)  =  -^^ — -, 

■written      "^  =    '^     ,   determines   two   values   for   tan  0. 

1  —  tan-  0      a  —  c 
Call  these  values  tan  6i  and  tan  Oo,  and  let  0^  locate  the  Xa-axis, 
$2  the  Fg-axis.     In   the   equation   axi^  +  2 bxiy^  +  cj/f  +k  =  0, 
substitute  Xj  =  r  cos  0,  ?/i  =  r  sin  0,  and  solve  for  ?-l     There  re- 
sults r'  =  -1c 'i-  +  t&^^'0 Calling  the  values  of  r 

a  +  2  &  tan  6^  -f  c  tan-  0 
corresponding  to  tan  Oi  and  tan  6.^  respectively  r^^  and  ?•2^  the 
equation  of  the  ellipse  or  hyperbola  referred  to  the  axes  X2,  Y^, 

IS     ^,-|---^;=l. 

rr     ?2- 

When  oc  —  b-  =  0,  the  general  ecjuation  becomes 

ax-  -f  2  a)(^xy  -f  cf  +  2dx  +  2  e>j  +  /  =  0, 
which  may  be  written  {a^x+chjf+2  dx-\-2  e//+f=().     Trans- 
form  to  rectangular  axes  with  a-.c  +  c-y  =  0  for  X-axis,  the 
origin  unchanged.     Then 

tan  6=~—   and    sin  6  =    ~  ^^'  ^,    cos  6  =  — ^— ^• 
c'  (a  +  cY  (a  +  c)-^ 

The  transformation  formulas  become 


(a  +  c)'^  (a  +  c)^ 


136  ANALYTIC   GEOMETllY 

The  transformed  equation  is 

2,0  ft-^  +  c^e  c)  ft-e  -  ckl     _       f 

Vi  +  -^ T  y^  — ' r  ^1  I' 

{a+cy  {a  +  cY  {a  +  cy 

which  may  be  written  in  the  form 

0/.-„y  =  2"'''~''y(.r,-m), 
the  equation  of  a  paraboLa  whose  parameter  is  2  — ^,  and 

(«  +  cy 

whose  vertex  referred  to  the  axes  Xj,  Y^,  is  {m,  n). 

The  condition  cfc  —  lr  =  0  causes  the  center  (a'o,  ?/o)  of  the 
conic  section  to  go  to  infinity.  Hence  the  parabola  may  be 
regarded  as  an  ellipse  or  hyperbola  with  center  at  infinity. 
When  the  discriminant  A  also  equals  zero,  the  parabola  be- 
comes two  straight  lines  intersecting  at  infinity ;  that  is,  two 
parallel  straight  lines. 

It  is  now  seen  that  every  second  degree  equation  in  two 
variables  interpreted  in  rectangular  coordinates  represents 
some  variety  of  conic  section.* 

Problems.  —  Determine  the  variety,  magnitude,  and  position  of  the 
conic  sections  represented  by  the  following  equations : 

1.    14  x2  -  4  xy  +  11  ?/2  -  44  X  -  58  2/  +  71  =  0. 

ac  —  h'=-\-  150,  therefore  the  equation  represents  an  ellipse.  The 
center  is  determined  by  the  equations 

14  xo  -  2  2/0  -  22  =  0,   -  2  xo  +  11  yo  -  29  =  0, 

*  The  three  varieties  of  curves  of  the  second  order  are  plane  sections  of 
a  right  circular  cone,  which  is  for  this  reason  called  a  cone  of  the  second 
order.  When  the  conic  section  becomes  two  parallel  straight  lines,  the 
cone  becomes  a  cylinder. 

Newton  (1642-1727)  discovered  that  the  curves  of  the  third  order  arc 
plane  sections  of  five  cones  which  have  for  bases  the  curves  21-25  on 
page  44.  Pliicker  (1801-18G8)  showed  that  curves  of  the  third  order 
have  219  varieties. 


SECOND    DEGUEE  EQUATIOy 


137 


to  be  the  point  (2,  3).  k  =  dxo  +  ei/o  +/ 
mined  in  direction  by  tan  (2  0)  =  —  j, 
wlience  2  tan"-  ^  -  3  tan  6-2  =  0,  tan 
=  2  or  -  J.  M+N=  25,  3IN  =  150. 
If  the  X-axis  corresponds  to  tan  6  =  2, 
M  —  Xniust  have  the  same  sign  as  b. 
Tiicrefore 

M  _  ,Y  =  _  5,    .1/  =  10,  .V  =  15. 
The  equation  of  tlie  ellipse 

10  a-2-  -h  15  2/2-  =  60, 


The  axes  are  deter- 


2/2- 


FiG.  125. 


2.   x:^  -  3  xij  +  2/-  +  10  X  -  10  2/  +  21  =  0. 

ac  —  b"  =  ~:l,  therefore  the  equation  represents  an  hyperbola.  The 
center,  determined  by  the  equations  Xo  —  %  ijo  +  5  =  0,  —  ^  xo  +  2/0  —  5=0, 
is  (—  2,  2).  A;  =  fZ.ro  +  eyo+f=+  \.  The  axes  are  determined  in  direc- 
tion by   tan  (2  0)  =00,   whence    0i=45°,    62 -IS^''-     By  substituting  in 

r^-  =  -k "^tJ^^ ,    ,,.  =  2,    r^  =  -l 

rt  +  2  ?*  tan  0  +  c  tan- 0 

The  equation  of  the  hyperbola  referred  to  its  own  axes  is  I  x-  —  ly-  =  I. 


Fm.  127 


3.   9  a:2  -  21  xij  +  16  2/2  -  18  x  -  101  y  +  19  =  0. 

rtc  —  62  —  0,  therefore  the  equation  represents  a  parabola.  Write  the 
equation  in  the  form  (3x-42/)2-18x-101 2/-M9=0.  Take  3.x-42/  =  0 
as  X-axis  of  a  rectangular  system  of  coordinates,  the  origin  unchanged. 
Then   tan  0  =  ?,    sin0=i?,    cos0  =  v;,    and  the  transformation   formulas 


138  ANALYTIC   GEOMETRY 

become    x  =  '^^^~^^S    y  =  §_^lJiAll.      The  transformed  equation  is 

5  5 

25?/i2-75a-i-70?/o+19  =  0,  wliicli  may  be  written  (?/i-|)2=3  (xi  +  g). 
Hence  the  parameter  of  the  parabola  is  3,  the  vertex  referred  to  the  new 
axes  (  —  I ,  I) . 

4.    2/2  +  2  a;y  +  3  x2  -  4  X  =  0.  5.    y^  +  2  xy  -  3  oc^  -  4  x  -  0. 

6.    ?/  -2xy  +  x^  +  x  =  0.  7.    y^  -2  xij  +  2  =  0. 

8.    ?/  +  4  x?/  +  4  x2  -  4  =  0.  9.    3  x2  +  2  xy  +  3y'^=  8. 

10.  4  x2  -  4  x?/  +  2/2  -  12  X  +  6  ?/  +  9  =  0. 

11.  x^  —  xy  —  6  2/2  =  6. 

12.  x^  +  xy  +  y-^  +  x  +  y  =  1. 

13.  3  x2  +  4  xy  +  2/2  -  3  X  -  2  ?/  +  21  =  0. 

14.  5  x2  +  4  X2/  +  2/"^  —  5  X  —  3  2/  —  19  =  0. 

15.  4  x2  +  4  X2/  +  2/-  -  5  X  -  2  2/  -  10  =  0. 

Art.  70.  —  Second  Degree  Equation  in  Oblique 
Coordinates 

To  determine  the  locus  represented  by 

ax'  +  2  hxy  +  c/  +  2  dx  +  2  c?/  +  /=  0,  (1) 

when  interpreted  in  oblique  axes  including  an  angle  ft,  let 

a'x"  +  2  6 'x'y'  +  cY'  +  2  rt'-^''  +  2e'y'+f'  =  0  (2) 

be  the  result  obtained  by  transforming  the  given  equation  to 
rectangular  axes,  the  origin  unchanged.  Since  (x,  y)  repre- 
sents any  point  P  referred  to  the  oblique  axes,-  and  {x\  y')  the 
same  point  referred  to  rectangular  axes,  the  expressions 

^  +  ?/  +  2  xy  cos  /3  and  x'-  +  y'^ 
are  each  the  square  of  the  distance  from  P  to  the  origin. 

Hence  x^  +  ?/-  +  2  xy  cos  ^  =  x'^  +  7j'\  (3) 

By  hypothesis 

ax'  +  2  &.^7/  +  c//-  =  «'•'«'-  +  2  6'.);'.v'  +  c^/'-.  (4) 

Multiply  the  identity  (3)  by  X  and  add  the  product  to  (4). 
There  results  the  identity 

(a  +  X)x^  +  2(b  +  \  cos  /3)  xy  +  (c  +  A)  ?/^ 

=  (a'  +  X)  x'2  +  2  b'x'y'  +  (c'  +  X)  y". 


SECOND  DEGREE  EQUATION  130 

Now  any  value  of  A  wliicli  makes  the  left-hand  member  of 
this  identity  a  perfect  square  must  also  make  tlie  ri,L,^ht-han(l 
memlier  a  perfect  square.     Tlic  left-hand  mcialjor  is  a  perfect 

,  n>  +  X  cos  BV     <■  +  X. 

S(iuare  when  /  — ^     =  — —  , 

\     a+X     J       a+X 

, ,    ,  •        ,        .  .>  ,  a  +c  —  2  6  cos  8  ^    ,  ac  —  b'-     ^ 

tliat  IS,  when  X-  -\ — — :-— ^^  X  -\ ; =  0. 

sin^  /3  sin-  /3 

The  riL,dit-hand  nieniher  is  a  perfect  square  when 

A-  +  0-t'  +  b')X  +  a'c'  -  b'-  =  0. 

Since  these  equations  determine  the  same  values  for  X, 

a'e'-b'-'  =  '-^^^^. 
sin-  /5 

Therefore  ac  —  b^  is  greater  than  zero  when  a'c'  —  b'-  is  greater 
than  zero.  When  a'c'  —  b''^  >  0,  equation  (2)  represents  an 
ellipse  when  interpreted  in  rectangular  coordinates.  Conse- 
quently when  ac  —  &-  >  0  equation  (1)  represents  an  ellipse 
when  interpreted  in  oblique  coordinates.  In  like  manner  it 
follows  that  equation  (1)  interpreted  in  oblique  coordinates 
represents  an  hyperbola  when  ac  —  Z^-  <  0,  a  parabola  when 
ac  -  62  =  0. 

Problems.  —  1.   Two  vertices  of  a  trianc;le  move  along  two  intersecting 
straight  lines.     Find  the  curve  traced  by  the  third  vertex. 
From  the  figure  are  obtained  the  pro- 

portionsof  y  =  ^'^Cg  +  °),  /y 

b  sin  oj 

X  _  sin  (e  +  CO  -  p) 
a  sin  w 

whence 


140 


ANALYTIC  GEOMETRY 


Substituting  in  sin^  d  +  cos-  ^  =  1,  there  results 


af  _  2  sin  (a  -  ;3  +  co)        _^  tf  ^  sin'-^  (a  +  ^  -  w) ^ 
a'^      "  ah  Ir  sin-  w 


tlie  equation  of  an  ellipse. 

2.  Find  the  envelope  of  a  straight  line  which  moves  in  such  a  manner 
that  the  sum  of  its  intercepts  on  two 
intersecting  straight  lines  is  constant. 

Let  -  +  -  =  1  be  the  moving  straight 
a      b 
line,  then  must  a -i-  b  =  c,  where  c  is  a 
constant.     The  equation  of  the  straight 

line  becomes  -  -| —  —  1,  which  may 

a  c  —  a 
be  written  a^ -\-(y  —  x  —  c)a  =  ex.  The 
equation  determines  for  every  point 
P(x,  y)  two  values  of  a,  to  which  cor- 
respond two  lines  of  the  system  inter- 
secting at  (x,  ?/).  When  these  two 
values  of  a  become  equal,  the  point  (;*•,  y)  becomes  the  intersection  of 
consecutive  positions  of   the  line;   that  is,  a  point  of  the   envelope  of 

the  line.  Hence  the  point 
(x,  y)  of  the  envelope  must 
satisfy  the  condition  that 
the  equation  in  a  has  equal 
roots.  The  equation  of  the 
envelope  is  therefore 

(y  —  X  —  c)2  +  4  c.^  =  0, 
which  reduces  to 
?/2  '-  2  xy  +  xr  —  2  cy  +  2  ex 

+  C2  =  0 

and  represents  a  parabola. 
This  problem  furnishes 
method  frequently  used 
to  construct  a  parabola  tan- 
gent to  two  given  straight  lines  at  points  equidistant  from  their  intersec- 
tion. Mark  on  the  lines  starting  at  their  intersection  the  equidistant 
points 
1,  2,  3,  4,  5,  G,  7,  8,  ••■,    -1,    -2,   -3,    -4,    -5,    -  G,    -7,    -8,  •••. 


SECOND   DEGREE  EQUATION 


141 


If  the  given  points  are  +  5  on  one  line  and  +  5  on  the  other,  the  straight 
lines  joining  the  points  of  the  given  lines  the  sum  of  whose  marks  is  +  5 
envelop  the  pai-abola  required. 

3.  Through  a  fixed  point  a  system  of  straight  lines  is  drawn.  Find 
the  locus  of  the  middle  points  of  the  segments  of  tliese  lines  includid  by 
the  axes  of  reference. 

4.  Find  the  envelope  of  a  straight  line  of  constant  lungtli  whose  ex- 
tremities slide  in  two  fixed  intersecting  straight  lines. 


Art.  71.  —  Conic  Section  through  Five  Points 


Let  (a-i,  ?/i),  (.«,,,  ?/,),  (a;,,,  ?/..j),  {x^,  y^)  be  four  points  of  which  no 
three  are  in  the  same  straight  line.  Let  a  =  0  be  the  straight 
line  through  (x^,  y{),  (x.,,  yS)]  b  =  0  the  line  through  (.i\,,  y.^, 
(xs,  2/3)  ;  c  =  0  the  line  through  (a%,  y.),  (x^,  y^  ;  d  =  0  the  line 
through  (a-4,  2/4),  O^'i,  Z/i)- 
The  equation  ac-\-'kbd=(), 
where  k  is  an  arbitrary 
constant,  represents  a 
conic  section  through  the 
four  points.  For,  since  a, 
b,  c,  d  are  linear,  the  equa- 
tion ac  +  kbd  =  0  is  of  the 
second  degree,  and  must 
therefore  represent  a  conic 

section.  The  equation  is  satisfied  by  a  =  0  and  b  =  0,  condi- 
tions which  determine  the  point  (ic^,  3/2) ;  by  a  =  0  and  d  —  0, 
determining  the  point  (xi,  y^);  by  c  =  0,  b  =  0,  determining 
(x.j,  1/3)  ;  by  c  =  0,  d  =  0,  determining  (a;^,  y^).  Since  k  is  arbi- 
trary, ac  +  kbd  =  0  represents  any  one  of  an  infinite  number 
of  conic  sections  through  the  four  given  points. 

If  the  conic  section  is  required  to  pass  through  a  fifth  point 
(x'5,  2/5)  not  in  the  same  straight  line  with  any  two  of  the  four 
points  {xi,  yy),  (x.,,  y^,  {x^,  y^),  (x^,  y^),  the  substitution  of  the 
coordinates  of  (x^,  2/5)  in  ac  +  kbd  =  0  determines  a  single  value 


142 


ANALYTIC  GEOMETRY 


for  Jc.     Therefore  five  points  of  wliich  no  three  lie  in  the  same 
straight  line  completely  determine  a  conic  section. 

Problems.  —  Find  the  equations  of  conic  sections  tlirough  the  five 
points. 

1.    (1,2),  (3,5),  (-1,4),  (-3,  -1),  (-4,  3). 

The  equations  of  the  sides  of  the  quadrilateral  whose  vertices  are 
the  first  four  points  are  a  =  Sx  —  2y  +  1  =0,  b  =  x  —  4tj  +  17  =  0, 
c  =  5 X  —  2  ?/  +  13  =  0,  d  =  Sx  —  4y  +  5  =  0.  The  equation  of  a  conic 
section  through  these  four  points  is  therefore 

(3x  -  2?/ +  l)(5x  -  2?/ +  13)+ i-(a;  -  4^  +  17)(3x  -  4?/ +  5)  =  0. 
Substituting  the  coordinates  of  the  fifth  point  (—4,  3),  k  =  W-     The 
equation  of  the  conic  section  through  the  five  points  is 

79  x2  -  320  xy  +  301  ?/2  +  noi  x  -  1665  y  +  1580  =  0. 

2.  (2,  3),  (0,  4),  (-  1,  5),  (-  2,  -  1),  (1,  -  2). 

3.  (1,  3),  (4,  -  G),  (0,  0),  (9,  -  9),  (16,  12). 

4.  (-  4,  -  2),  (2,  1),  (-6,  3),  (0,  0),  (2,  -  1). 

5.  (-  i,  -  i),  (2,  1),  (f,  2),  (-J,  -  3),  (I,-  I). 

6.  (3,  V5),  (-2,  0),  (-4,  -  Vl2),  (3,  -  V5)  (2,  0). 

7.  (1,2),  (2,  1),  (3,  -2),  (0,4),  (3,0). 

8.  (2,3),  (-2,3),  (4,1),  (1,3),   (0,0). 


Art.  72.  —  Conic  Sections  Tangent  to  Given  Lines 

Let  «  =  0  and  b  =  0  represent  two  straight  lines  intersected 
hy  the  straight  line  c  =  0.  The  equation  ab  —  kc'  =  0  repre- 
sents a  conic  section  tan- 
gent to  the  lines  a  =  0, 
6  =  0  at  the  points  of  inter- 
section of  c  —  0.  For  the 
equation  ab  —  kc'  =  0  is  of 
the  second  degree,  and  the 
points  of  intersection  of  the 
line  a  =  0  with  ab  —  kc^  =  0 
^'"^  ''^^'  coincide  at  the  point  of  in- 

tersection of  the  lines  a  =  0,  c  =  0,  which  makes  a  =  0  tangent 
to  the  conic  section.     For  a  like  reason  b  =  0  is  tangent  to 


SECOND   DEGliEK  EQUATION 


143 


ab  —  kc-  —  0.  Since  k  is  arbitrary,  an  infinite  number  of  conic 
sections  can  be  drawn  tangent  to  the  given  lines  at  the  given 
points. 

The  equation  of  a  conic  section  tangent  to  the  lines  x  =  (), 
y  =  0  lit  the  i)oints  (a,  0),  (0,  h)  is 


a      b 


Kxy  =  0. 


(1) 


The  points  of  intersection  of  this  conic  section  and  the  line 

MK  -  +  ^  =  1,    lie    in    the 
m      11 

locus  of  the  equation 

i?  +  f_5_?'Y  =  A>,.  (2) 
a      b     m     nj 

This  last  equation  is  homo- 
geneous of  the  second  degree, 
and  hence  represents  two 
straight  lines  from  the  origin 
through  the  points  of  inter- 

r       a"     ,    2/         1         A  Fig.  138. 

section     of [--—1  =  0 

VI      n 

and  (-  +  -  —  1)  —  Kxy  =  0.    The  straight  lines  represented  by 


equation  (2)  coincide,  and  -    +  -  —  1  =  0  is  tangent  to 


when 

is  a  perfect  square ;  that  is,  whei 


a      b         ' 

I      b      m      -  '  ^ 


\rt      mj  \b      71 J        I  \a      mj  \b      nj       2  S 


144  ANALYTIC  GEOMETRY 

whence  K=  if^-  lY/^i  -  ^Y  (3) 

\(i      iiij  \h      nj 

Similarly,  —  +  -  —  1  =  0  is  tangent  to  (1)  when 


Equations  (3)  and  (4)  determine  the  values  of  -  and  -  in  terms 

of  the  arbitrary  constant  K,  which  shows  that  an  infinite  num- 
ber of  conic  sections  can  be  drawn  tangent  to  four  straight 
lines  no  three  of  which  pass  through  a  common  point.     If 

[-  — =  1  is  also  tangent  to  the  conic  section  represented  by 

equation  (1), 

A-=4fl^i)fl-l>  (5) 

Equations  (3),  (4),  (5)  determine  -,  -,aud  A"  uniquely,  proving 

a  h 
that  only  one  conic  section  can  be  drawn  tangent  to  five  straight 
lines  no  three  of  which  pass  through  a  common  point.  This 
proposition  is  the  reciprocal  of  the  j)roposition  of  Art.  71  and 
might  have  been  demonstrated  by  the  method  of  reciprocal 
polars. 

Problems.  —  1 .  Find  the  equation  of  the  parabola  tangent  to  two 
straight  lines  including  an  angle  of  60°  at  points  whose  distances  from 
their  point  of  intersection  are  2  and  4. 

2.  Find  the  equation  of  the  conic  section  tangent  to  two  straight  lines 
including  an  angle  of  45°  at  (3,  0),  (5,  0),  and  containing  the  point  (7,  8), 
the  given  straight  lines  being  the  axes  of  reference. 


Art.  73,  —  Similar  Coxic  Sections 

The  points  P{x,  y)  and  P^iinx,  my)  lie  in  the  same  straight 
line  through  the  origin  0,  and  0I\  =  m  •  OP.  The  distance 
between  any  two  positions  of  P^,  (mx',  my'),  (mx",  my")  is  m 


SECOND  DEGREE  EQUATION 


145 


times  tlie  distance  between  tlie  correspondiut,^  positions  of 
P,{x\y'),{x'\y").     For 

{ (m.f'  -  mx")-  +  {my'  -  myy\  ^  =  m  \  (^x'  -  x'J  +  (//'  -  y"f\  i 

Representing  the  point  P  by  {x,  y),  the  point  P^  ])y  (X,  Y), 
when  {x,  y)  traces  a  geometric  figure,  the  point  (A",  Y)  traces  a 
figure  to  scale  m  times  as  large.     The  effect  of  tlie  substitution 

X  Y  . 

X——,  ?/  =  —  is  therefore  simijly  to  change  the  scale  of  tlu; 

drawing.     Figures  thus  related  are  said  to  be  similar.     When 

the  two  equations /(;r,  y)  =  0,  fi~-,  —  j  =  0  are  interpreted  in 

the  same  axes,  their  loci  are  similar  and  similarly  placed ; 

when  interpreted  in  differ-  ^ 

ent  axes  but  including  the 

same  angle,  the  loci   are 

similar.     Ellipses  similar 

to  —  +  -i-  =  1    are    repre- 

sented  by  Al  +  ^      1. 


All  ellipses 

of  a 

similar 

system  have 

the  same  ec- 

centricity,  for 

«-^ 

r  -  7;* 
m-a^ 

^ 

_  a-  - 

-b\ 

Y2 


In  like  manner,  all  hyperbolas  of  a  similar  system  -^  • 

??i^a^     m'b- 
have  the  same  eccentricity.     The  parabolas  similar  to  y^  =  2x)x 
are  represented  by  the  equation  Y^  =  22)mX. 
Taking  as  corresponding  points 


J'(^,  y),     Pi(mx,  my),     !'.,(-  mx, 


'Z/), 


146 


ANALYTIC  GEOMETRY 


the  figure  traced  by  Pi  is  similar  to  that  traced  by  P,  the  figure 
traced  by  P2  is  symmetrical  to  that  traced  by  P^. 

The  change  of  scale  of  a  drawing  may  be  effected  mechani- 
cally by  means  of  an  instrument  called  the  pantograph,  which 
consists  of  four  rods  jointed  together  in  such  a  manner  as  to 
form  a  parallelogram  ABOC  with  sides  of  constant  length, 
but  whose  angles  may  be  changed  with  perfect  freedom.  On 
the  rods  AB  and  AC  fix  two  points  P  and  Pj  in  a  straight 
line  with  0.  If  the  point  0  is  fixed  in  the  plane,  and  the 
point  P  is  made  to  take  any  new  position  P',  and  the  cor- 
responding position  of  Pj  is  P/,  the  points  P',  0,  P/  in  Fig. 
134  are  always  in  a  straight  line,  the  triangles  Pi  CO  and 

Pi'A'P'   are  similar   and 
'  P  hence 


qpi 

OP' 


P,'C 

CA' 


m 

CA 


a  constant  which  may  be 
denoted  by  m.  Taking  0 
as  origin  of  a  system  of 
rectangular  coordinates, 
if  Pis  (a-,  2/),  Pi  is 

(—  mx,  —  my). 

If  the  point  P  is  fixed  in 
the  plane  and  taken  as  origin  of  a  system  of  rectangular  coordi- 
nates, if  the  point  0  is  (x,  y),  the  point  Pj  is  {mx,  my).  There- 
fore, if  the  point  0  is  made  to  trace  any  locus,  the  point  P, 
traces  a  similar  figure  to  a  scale  m  times  as  large. 


The  equation 


Art.  74.  —  Confocal  Conic  Sections 
1, 


If 


(1) 


a^  -f-  A      6'  -f  A 

where  a?  >  W  represents  an  ellipse  when  A  >  —  6-,  an  hyperbola 
when  —  a^  <  A  <  —  &^,  an  imaginary  locus  when  A  <  —  al    The 


SECOND   DEGREE  EQUATION 


147 


distance  from  focus  to  center  of  the  ellii)se,s  and  liyperbolas 
represented  by  eqiiation  (1)  is  \(i' +  \  — Ir —  \\'- =  {tC- —  U')-. 
Hence  equation  (1)  when  interpreted  for  different  values  of  A. 
in  the  same  rectangular  axes  represents  ellipses  and  hyperbolas 
having  common  foci ;  that  is,  a  system  of  confocal  conic  sections. 
Through  every  point  {x',  y')  of  the  plane  there  passes  one 
ellipse  and  one  hyperbola  of  the  confocal  system 

a^  +  A      h'  +  X 
For  the  conic  sections  passing  through  {x\  y')  corresptmd  to 
the  values  of  X  satisfying  the  equation 


.. 


1.     (2) 


a-  +  X      b-  +  X 
This  function  of  A, 

.^1^  +  ^ 1, 

a-  +  A      h-  +  X 

is  negative  when  A  =  +  co, 
positive  just  before  A  be- 
comes —  6-,  negative  when 
A  is  just  less  than  —  li' 
and  again  positive  when 
A  is  just  greater  than  —  cr. 
Hence  equation  (2)  deter- 
mines for  A  two  values,  one  between  +  x  and  —  h'-,  the  other 
between  —  Jr  and  —  a'. 

The  ellipse  and  hyperl)o]a  of  the  confocal  system 

(t-  -H  A      //-  -f  A 
til  rough  the  point  (.c',  //')  intersect  at  right  angles. 

Let  Ai  and  A.  be  the  values  of  A  satisfying  the  equation 


X' 


+ 


a-  +  A      h-  +  X 


=  1.     Then 


+ 


a-  +  Ai      &'  -f  Ai 


a-  +  X.,     U-  +  A. 


148  ANALYTIC  GEOMETRY 

represent  ellipse  and   hyperbola  tlirougli    (x-',  y').     The  tan- 
gents to  this  ellipse  and  hyperbola  at  (x',  y')  are 


iiy 


a"  +  Ai      h-  +  Ai 

xx'  yy' 

a^  +  A2     b^  +  A2 


(1) 
(2) 


From  the  equations 


+  r7^^=l.        -.-^^  + 


a^  +  Ai      b-  +  \i  a-  +  \2      b'' +  X., 

is  obtained  by  subtraction 


(cr  +  Ai)  (a'  +  \,)      {b'  +  Aj)  (6^  +  A.) 

which  is  the  condition  of  perpendicularity  of  tangents  (1) 
and  (2). 

Since  through  every  point  in  the  plane  there  passes  one  ellipse 
and  one  hyperbola  of  the  confocal  system,  the  point  of  the 
plane  is  determined  by  specifying  the  ellipse  and  hyperbola  in 
which  the  point  lies.  This  leads  to  a  system  of  elliptic  coordi- 
nates. 

If  heat  flows  into  an  infinite  plane  disc  along  a  finite  straight 
line  at  a  uniform  rate,  when  the  heat  conditions  have  become 
permanent,  the  isothermal  lines  are  the  ellipses,  the  lines  of 
flow  of  heat  the  hyperbolas  of  the  confocal  system.  The  same 
is  true  if  instead  of  heat  any  fluid  flows  over  the  disc,  or  if  an 
electric  or  magnetic  disturbance  enters  along  the  straight  line. 


CHAPTER   XI 
LINE  OOOKDINATES 

Art.  75.  —  Coordinates  of  a  Straight  Line 

If  the  equation  of  a  straight  line  is  written  in  the  form 
ux  -\-vy  +  1  =  0,  u  and  v  are  the  negative  reciprocals  of  the 
intercepts  of  the  line  on  the  axes.  To  every  pair  of  values  of 
H  and  V  there  corresponds  one  straight  line,  and  conversely;  that 
is,  there  is  a  "  one-to-one  correspondence  "  between  the  symbol 
(«,  v)  and  the  straight  lines  of  the  plane,  u  and  v  are  called 
line  coordinates.* 

If  {u,  V)  is  fixed,  the  equation  ux -{- vy  -\- 1  =  0  expresses  the 
condition  that  the  point  {x,  y)  lies  in  the  straight  line  («,  v). 
The  system  of  points  on  a  straight  line  is  called  a  range  of 
points.  Hence  a  first  degree  point  equation  represents  a  range 
of  points  and  determines  a  straight  line. 

If  (x,  y)  is  fixed,  ux  +  vy  +  1  =  0  expresses  the  condition  that 
the  line  (»,  v)  passes  through  the  point  (x,  y).  The  system  of 
lines  through  a  point  is  called  a  x^encil  of  rays.  Hence  a  first 
degree  line  equation  represents  a  pencil  of  rays  and  determines 
a  point. 

The  equations  ?/.ri  +  vy^  -|-  1  =  0,  nx^  +  vy.,  +  1=0  determine 
the  points  {x^,  y,),  {x^,  y^  respectively.  \\\^'^  '\+^) 
represents  for  each  value  of  X  one  point  of  the  line  through 
(a^i,  ?/i),  (.i\,,  7/2).  X  is  the  ratio  of  the  segments  into  Avhich  the 
point  corresponding  to  X  divides  the  finite  line  from  (.r„  y,)  to 

*  riiickcr  in  Germany  and  Cliasles  in  France  developed  the  use  of  line 
coordinates  at  about  the  same  time  (1829). 
149 


150  ANALYTIC  GEOMETUr 

{x2,  2/2)-  There  is  a  "  one-to-one  correspondence "  between  A 
and  tlie  points  of  the  line  throngh  (a-j,  y^,  (.i-^,  y^. 

1-f-A  1  +  A 

which  reduces  to  {ux^  +  vy^  +  1)  -|-  A  {ax^  +  vy2  +  1)  =  0,  is  the 
line  equation  of  the  point  A.  Denoting  m.Ti  +  vy^  -f  1  by  L, 
UX2  +  v?/2  +  1  by  3f,  L  +  \3I—  0  represents  the  range  of  points 
determined  hy  L  =  0,  3£—  0. 

The  rays  of  the  pencil  determined  by  the  lines 

UiX  +  Viy  +  1  =  0,  ti.^x  -\-  v^y  +  1  =  0 

are  represented  by  the  equation 

(u,x  -f  v,y  +  1)  +  A  (n.x  +  v-^y  +  1)  =  0, 

which  may  be  written  '^!l±^x  +  !!l±J^  +  1  =  0. 
•^  1+A  1  +  A 

Hence  (n,  +  Xv     v,±M:: 

V  1+A  1+A 
are  the  lines  of  the  pencil  determined  by  (11^,  Vi),  (xi2,  v^).  There 
is  a  "  one-to-one  correspondence  "  between  A  and  the  rays  of 
the  pencil.  Denoting  UiX  -[■  Viy  -\-l  by  P,  u^x  +  Vjy  +  1  by  Q, 
P  +  AQ  =  0  represents  the  rays  of  the  pencil  determined  by 
P=0,  Q  =  0.  ' 

Problems.  —  1.    Construct  the  lines  (4,  1);  (-  2,  5);  (-  i,  -  J). 

2.  Construct  the  pencil  represented  by  3  ?i  -  2  y  ^-  1  =  0. 

3.  Construct  the  range  represented  by  2  .x  —  3  ?/  -f  1  =  0. 

4.  Locate  the  point  determined  by  4  ?(  -^  5  v  -|-  1  =  0. 

5.  Draw  the  line  determined  by  3x  —  5  ?/  +  1  =  0. 

6.  Write  the  equation  of  the  range  of  points  determined  by 

2ti-Sv  +  l  =  0,  i?t-f^w-fl=0. 

7.  Write  the  equation  of  the  pencil  of  rays  determined  by 

2x-Sy  +  l=0,  J  a;-f  i?/  +  l  =  0. 


LINE  COORDINATES  151 


Akt.  76.  —  Line  Equatioxs  of  the  Conic  Sections 

The  equation  of  the  tangent  to  the  ellipse   ^—■^•-L^—l  at 

a-      b- 
(.T,,,  y/o)  is  ^'  +  "''•'^  =  1.    Comparing  the  equation  of  the  tangent 

a-        U- 
with  nx  +  (•>)  +  1  =  0  it  is  seen  that  the  line  coordinates  of  the 

tangent  are  a  =  —  ' ",  i'  =  —  ^",  whence  a-,,  =  —  u-u,  ?/„  =  —  Irv. 
If  the  point  of  tangency  (.Tn,  ?/„)  generates  the  ellipse  "^  +  ii  —  •'-> 
the  tangent  {ii,  v)  envelopes  the  ellipse.  Hence  the  line  equa- 
tion of  the  ellipse,  when  the  reference  axes  are  the  axes  of  the 
ellipse,  is  a-ir  +  b'-v^  —  1. 

Problems.  —  1.    Show  that  the  Hue  equation  of  the  circle  x"^  +  y-  —  r" 

is  iC-  +  y-  =  — 
r- 

2.    Show  that  the  Hue  eiiuatiou  of  the  hyperbola 


1  is  d-xi-  —  b-v-  =  1. 


a-     h- 

3.  Show  that  the  Hue  equatiou  of  the  parabola  y"^  =  2px  is  pv'^  =  2  u. 
Construct  the  euvelopes  of  the  equations 

4.  -  +  -  =  -  5.  G.    ifi-\-  v"^  =  '  8.   9  1*2  -  4  u2  =  i. 

U        V 

5.  uv-\.  7.    9!t2  +  4tj2-i,  9.    8i;2-u  =  0. 


Art.  77.  —  Cross-ratio  of  Four  Points 

The  double  ratio  -^--. — ^  is  called  the  cross-ratio  of  the  four 
CB     I)B 
points  A,  B,  C,  D,  and  is  denoted  by  the  symbol  (ABCB).     If 
the    point   A   is    denoted   by 

i  =  0,  the  point  B  by  J/=  0,    > ^ »< ^ 

the  points   C  and   D  respec-  ^'"-  '"■ 

tively   by    L  +  XiM=Q   and   L  +  XoM^O,    it    follows   that 

^  =  Ai,  ^  =  X,,,  and  (ABCD)  =  ^-     Take  any  four  points  of 
CB  JJB  Ao 


152  ANALYTIC   GEOMETRY 

tlie  range  L  +  X3/=  0  corresponding  to  Aj,  X.,,  A3,  A4,  and  repre- 
sent L  +  Aji)/  by  Li,  L  +  LM  by  i»/i,  whence   /v  +  A,,.1/  is 

represented  by  L,-^^^M„  L  +  A, J/   by    L,  -  ^i^il/-i. 

Ao  —  A3  Ao  —  A4 

The  four  points  corresponding  to  Aj,  A.,  A3,  A4  are  represented 
by  the  equations 

L,  =  0,  3/1  =  0,  A  -  ^^^^^^3/,  =  0,  A  -  ^i-^^1A  =  0, 
A2  —  A3  A2  —  A4 

and  their  cross-ratio  is  ^^  ~    ^    -~    ^     Since  the  four  points 

^2  —  X^Xi  —  A4 

Ai,  Ao,  Ag,  A4  can  be  arranged  in  24  different  ways,  the  cross- 
ratio  of  four  points  takes  24  different  forms,  but  these  24 
different  forms  are  seen  to  have  only  six  different  vaUies. 

i-f  A3/=0,  L'  +  XM'  =  0  represent  two  ranges  of  points. 
By  making  the  point  of  one  range  determined  by  a  value  of 
A  correspond  to  the  point  of  the  other  range  determined  by  the 
same  value  of  A,  a  "  one-to-one  correspondence  "  is  established 
between  the  points  of  the  two  ranges,  and  the  cross-ratio  of 
any  four  points  of  one  range  equals  the  cross-ratio  of  the  corre- 
sponding four  points  of  the  second  range.  Such  ranges  are 
called  projective. 


Art.  78.  —  Second  Degkke  Line  Equations 

Remembering  that  each  of  the  equations 

L  +  \M=0,         i'-fAJ/'  =  0 

for  any  value  of  A  represents  the  entire  pencil  of  rays  through 
the  point  of  the  range  corresponding  to  A,  it  is  evident  that  the 
equation  LM'  —  L'M=  0,  obtained  by  eliminating  A  between 
L  +  \M=^  0,  L'  +  AIT'  =  0,  represents  the  system  of  lines  join- 
ing the  corresponding  points  of  the  two  projective  point  ranges. 
This  equation  is  a  second  degree  line  equation,  and  it  becomes 
necessary  to  determine  the  locus  enveloped  by  the  lines  repre- 
sented by  the  equation. 


LINE   COOliDINATES  153 

Let  nx  +  vy  +  1  =  0  represent  any  i^oint  (x,  y)  of  the  plane. 
Writing  the  values  of  L,  M,  L',  31'  in  full,  the  elimination  of 
u  and  V  from  the  equations  i(x  +  t'^  +  1  =  0, 

iixi  +  viji  +  1  +  X(ux2  +  vyo  +  1)  =  0? 
ux'  +  vy'  +  1  +  X(ux"  +  vy"  +  1)  =  0, 

determines  a  quadratic  equation  in  X  with  coefficients  of  the 
first  degree  in  (x,  ?/),  GX^  +  HX  +  K=  0.  To  the  two  values 
of  X  which  satisfy  this  equation  there  correspond  the  tangents 
from  (x,  y)  to  the  envelope  of  LM'  —  VM  —  0.  When  these 
tangents  coincide,  the  point  (.r,  ?/)  lies  on  the  envelope. 

4  IP  -  GK^  0 

causes  the  coincidence  of  the  tangents,  and  is  therefore  the 
point  equation  of  the  envelope.  The  point  equation  being  of 
the  second  degree,  the  envelope  is  a  conic  section. 

The  degree  of  a  line  equation  denotes  the  number  of  tan- 
gents that  can  be  drawn  from  any  point  in  the  plane  to  the 
curve  represented  by  the  equation,  and  is  called  the  class  of 
the  curve. 


Art.  79.  —  Cross-ratio  of  a  Pencil  of  Four  Eays 

Let  a  pencil  of  four  rays, 

P=0,      Q  =  0,      P-|-X,Q  =  0,      P-fA,Q  =  0, 

be  cut  by  any  transversal  in  the  four  points  A,  B,  C,  D.  j^  is 
the  common  altitude  of  the  triangles  whose  common  vertex 
is  0,  and  whose  bases  lie  in  the  transversal.     Then 

i>  •  CA  =  OA  -  OC  •  sin  COA,  p  •  DA  =  OA  ■  OD  ■  sin  DO  A, 
p -03=00 -OB-  sin  COB,  p  -  DB  =  OD  •  OB  ■  sin  DOB, 

and  (ABCD)  =  ^"^  ^^^^  -  ^"^^^^^.     This  double  sine  ratio  is 
^  ^      sin  COB      sin  DOB 


154  ANALYTIC   GEOMETRY 

called  the  cross-ratio  of  the  pencil  of  four  rays.     It  is  evident 

that  central  projection  does 
not  alter  the  cross-ratio  of 
four  points   in   a  straight 
'PA    \      "^^  line. 

Writing     the     equation 
P-|-XQ  =  0    in    the    com- 
Q=0  plete  form 
u^x  +  i\y  +  1 

FiG'  138.  and  this  in  the  form 

the  factor  ^ll^ll+i^iX  is  seen  to  be  the  negative  ratio  of  the 

distances  from  any  point  of  the  line  P  +  \Q  =  0  to  the  lines 

P  =  0,  Q  =  0.     Hence 

(7a_sinC0^1__.  Da'  ^^mPOA^     ^ 

Cb      sin  COB  "         Db'      sin  DOB  " 

^  sinCO^^sin^DOA^Xi^^j^g  cross-ratio  of  the  four  rays 
sin  COB     sin  X)0i5     Xg 

P^O,      Q  =  0,    P4-AiQ  =  0,    P  +  X,Q  =  0. 

Representing 

P+X,Q   by    Pi,     P  +  XoQ  by    Q„     P-f-XgQ 

is  represented  by 

P^  _  k^A^  Q„     p  +  X.Q  by    P.  -  ^^  Qi. 

Xa  —  X3  Ao  —  A4 

Hence  the  cross-ratio  of  the  four  points  of  the  pencil  P+XQ—0 

, .        ,      ,      -      ,      1     •     Xi  —  Xi  Xo  —  X) 

corresponding  to  Xj,  Xo,  X;,,  A4  is -^ — • 

X2  —  A.3  Aj^  • —  A4 


LINE  COORDINATES  155 

By  making  the  ray  of  F+kQ  =  0  determined  by  a  value  of 
\  correspond  to  the  ray  of  F'  +XQ'  =  0  determined  by  the 
same  value  of  X,  a  "  one-to-one  correspondence  "  is  established 
between  the  rays  of  the  two  pencils,  and  the  cross-ratio  of  any 
four  rays  of  one  pencil  equals  the  cross-ratio  of  the  correspond- 
ing four  rays  of  the  other  pencil.  Such  pencils  are  called 
projective  pencils. 

The  equation  of  the  locus  of  the  points  of  intersection  of  the 
corresponding  rays  of  the  two  projective  pencils  F  +  XQ  =  0, 
F'  -t-  XQ'  =  0  is  FQ'  —  F'Q  =  0.  This  is  a  second  degree  point 
equation  and  represents  a  conic  section.* 


Art.  80.  —  Construction"  of  Projective  Ranges  and 
Pencils 

If  there  exists  a  "  one-to-one  correspondence  "  between  the 
points  of  two  ranges,  between  the  rays  of  two  pencils,  or  be- 
tween the  points  of  a  range  and  the  rays  of  a  pencil,  the  ranges 
and  pencils  are  projective. 

Let  F=0,  Q  =  0,  determining  the  range  or  pencil  F-}-XQ=0, 
correspond  to  F^  =  0,  Qi  =  0,  determining  the  range  or  pencil 
Pi+ A,Qi  =  0,  and  let  a  "one-to-one  correspondence"  exist 
between  the  elements  X  of  the  first  system  and  the  elements 
Ai  of  the  second  system.  This  "one-to-one  correspondence" 
interpreted  algebraically  means  that  Xi  is  a  linear  function 

of  X;  that  is,  Xj  =  ^Jhj±A.     By  hypothesis,  Xi  =  0  when  X  =  0, 

'    '      cX  +  d         ^     ^^  ^    ' 

and    X,  =  CO    when   X  =  cc,    hence    b  =  0,    c  =  0,    and    X,  =  -  X. 

Let  X  =  /   and  X,  =  /,  be  a  third   ])air  of  corresponding  ele- 

*  A  complete  projective  treatment  of  conic  sections  is  developed  in 
Steiner's  Theorie  der  Kcgelschnitte,  1800,  and  in  Chasles'  G^om^trie 
Sup^rieure,  1852,  and  in  Cremona's  Elements  of  Projective  Geometry, 
translated  from  the  Italian. 


156 


ANALYTIC  GEOMETRY 


ments;tlien  -  =  -,  Ai  =  -X,  and  the  equations  of  tlie  systems 

P+XQ  =  0,  Pi  +  Ai(3i  =  0  become  P  +  XQ  =  0,    IP^  +  XI,Q,^0. 

Now  the  elements  of 
^  IF,  +  XIQ,  =  0 

are  the  elements  of 

P,  +  XQ,  =  0, 
hence  the  systems  between 
whose  elements  there  exists  a 
'^  one-to-one  correspondence  " 
are  the  projective  systems 
Fm.  m  P-{-XQ  =  0,    Pi  +  AQ,  =  0. 

This  analysis  also  shows  that  the  correspondence  of  three  ele- 
ments of  one  system  to  three  elements  of  another  makes  the 
systems  projective. 

Projective  systems  are  constructed  geometrically,  as  follows : 
Let  the  points  1,  2,  3  on  one  straight  line  mm  correspond  to 

the  points  1,  2,  3,  respec- 
tively, on  another  straight 
line  nn.  Place  the  two 
lines  with  one  pair  of 
corresponding  points  2,  2 
in  coincidence.  Join  the 
point  of  intersection  0  of 
the  lines  through  1, 1  and 
3,  3  with  2.  Take  the 
3  points  of  intersection  of 
lines  through  0  with  mm 
and  7in  as  corresponding 
points,  and  a  "  one-to-one  correspondence  "  is  established  between 
the  points  of  the  ranges  mm,  nn,  which  are  therefore  projective. 
In  like  manner,  if  three  rays  1,  2,  3  of  pencil  m  correspond 
to  the  rays  1,  2,  3,  respectively,  of  pencil  w,  by  placing  the  cor- 
responding rays  1,  1  in  coincidence,  and  drawing  the  line  00 


LINE  COORDINATES 


157 


through  the  points  of  intersection  of  the  corresponding  rays 
2,  2  and  3,  3,  and  taking  rays  from  m  and  7i  to  any  point  of  00 
as  corresponding  rays,  a  "  one-to-one  correspondence  "  is  estalv 
lished  between  the  rays  of  the  two  pencils,  and  the  pencils  are 
projective. 

Art.  81.  —  Coxic  Section  through  Five  Points 

It  is  now  possible  by  the  aid  of  the  ruler  only  to  construct  a 
conic  section  through  five  points  or  tangent  to  five  lines.  Take 
two  of  the  given  points  1,  2  as  the  vertices  of  pencils,  the  pairs 
of  lines  from  1  and  2  to  the  remaining  three  points  3,  4,  5, 
respectively,  as  corresponding  rays  of  projective  pencils.     The 


2 

/^ 

7 

1(5)/ 

/ 

). 

y 

(7) 

(3^ 

3 

line  11  is  a  transversal  of  pencil  1,  22  of  pencil  2.  0,  the 
intersection  of  51  and  32,  is  the  vertex  of  a  pencil  of  which 
11  and  22  are  transversals.  Hence  the  pencils  1  and  2  arc 
projective,  and  corresponding  rays  are  rays  to  the  points  of 
intersection  of  the  rays  of  pencil  0  with  11  and  22.     The 


158 


ANALYTIC  GEOMETRY 


intersections  of  these  corresponding  rays  are  points  of  the 
required  conic  section. 

Take  tAvo  of  the  five  given  lines  11  and  22  as  bearers  of  point 
ranges  on  which  the  points  of  intersection  of  the  other  lines 


33,  44,  55  respectively  are  corresponding  points.  The  line  00 
is  a  common  transversal  of  the  pencils  (11),  (22).  Hence 
corresponding  points  of  the  projective  ranges  11  and  22  are 
located  by  the  intersection  with  11  and  22  of  lines  connecting 
(11)  and  (22)  respectively  with  any  point  of  00.  The  straight 
lines  connecting  corresponding  points  are  tangents  to  the 
required  conic  section. 

Notice  that  the  construction  of  the  conic  section  tangent  to 
five  straight  lines  is  the  exact  reciprocal  of  the  construction  of 
the  conic  section  through  five  points. 

The  figure  formed  by  joining  by  straight  lines  six  arbitrary 
points  on  a  conic  section  in  any  order  whatever  is  called  a  six- 


LINE   COORDINATES 


ir)9 


7-f-\!(2)\4 


.''5 


point  Taking  1  and  5  as  vertices  of  pencils  whose  correspond- 
ing rays  are  determined  by  the  points  2,  3,  4,  the  points  of 
intersection  of  16  with  11  and  of  5iy  with  55  must  lie  in  the 
same  ray  of  the  auxiliary  pencil  0;  that  is,  in  any  six-point  of 
a   conic  section  the   inter-  ^i 

section  of  the  three  pairs  \    1\ 

of  opposite  sides  are  in 
a  straight  line.  This  is 
Pascal's   theorem.* 

Reciprocating  Pascal's 
theorem,  Brianchon's  theo- 
rem is  obtained.  —  In  the 
figure  formed  by  drawing 
tangents  to  a  conic  section 
at  six   arbitrary  points  in  fig.  u3. 

any  order  whatever  (a  six-side  of  a  conic  section),  the  straight 
lines  joining  the  three  pairs  of  opposite  vertices  pass  through 
a  common  point.f 

By  Pascal's  theorem  any  number  of  points  on  a  conic  section 
through  five  points  may  be  located  by  the  aid  of  the  ruler ;  by 
Brianchon's  theorem  any  number  of  tangents  to  a  conic  section 
tangent  to  five  straight  lines  may  be  drawn  by  the  aid  of  the 
ruler. 

*  Discovered  by  Pascal,  1040. 
t  Discovered  by  Brianclion,  180G. 


CHAPTER   XII 
ANALYTIC   GEOMETRY   OP   THE  COMPLEX   VAEIABLE 

Art.  82. —  Graphic  Represextation  op  the  Complex 
Variable 

The  expression  x  +  iij,  where  x  and  y  are  real  variables  and 
i  stands  for  V—  1,  is  called  the  complex  variable,  and  is  fre- 
quently represented  by  z.  Vx'  +  if  is  called  the  absolute  value 
of  z  and  is  denoted  by  1 2;  |  or  |  a;  +  iy  |. 

If  a;  +  iy  is  represented  by  the  point  (x,  y),  a  "  one-to-one 
correspondence"  is  established  be- 
^x-^iy  tween  the  complex  variable  x  +  iy 

j  and  the  points  of  the  XF-plane. 

[  The  X-axis  is  called  the  axis  of 

X       \ reals,  the  F-axis  the  axis  of  imagi- 

^    naries.    Denoting  the  polar  coordi- 

FiG.  144.  nates  of  (x,  y)  by  r  and  6,  x=  r  cos  6, 

y  =  r  sine,  and  2=x-+ «/=?•  (cos  ^+r  sin ^),  where  r=Vo^+f, 

6  =  tan-^^.     r  is  the  absolute  value,  and  6  is  called  the  anipli- 

X 

tude  of  the  complex  variable  x  +  iy.  Hence  to  the  complex 
variable  x  +  iy  there  corresponds  a  straight  line  determinate 
in  length  and  direction.  A  straight  line  determinate  in  length 
and  direction  is  called  a  vector.  Hence  there  is  a  "  one-to-one 
correspondence"  between  the  complex  variable  and  plane 
vectors.  As  geometric  representative  of  the  complex  variable 
may  be  taken  either  the  point  (x,  y)  or  the  vector  which  deter- 
mines the  position  of  that  point  with  respect  to  the  origin.* 

*  Argand  (1806)  was  the  first  to  represent  the  complex  variable  by 
points  in  a  plane.     Gauss  (1831)  developed  the  same  idea  and  secured 
for  it  a  permanent  i>lace  in  mathematics. 
160 


COMPLEX    VAIIIABLE 


161 


Calling  a  liiu^  etiual  in  Icnii^'tli  to  the  linear  unit  and  laid  olF 
from  the  origin  along  the  positive  direction  oi'  the  axis  of  reals 
the  uuit  vector,  the  complex  variable 

z  —  X  -\-  i>i  —  r  (cos  6  +  I  sin  6) 

represents  a  vector  obtained  by  multiplying  the  unit  vector  by 
the  absolute  value,  then  turning  the  resulting  line  about  its  ex- 
tremity at  the  origin  through  an  angle  equal  to  the  amplitude  of 
the  complex  variable.  When  the 
complex  variable  is  written  in 
the  form  r(cos  ^  +  «  sin(9),  r  is 
the  length  of  the  vector, 

cos  6  -}-  i  sin  9 

the  turning  factor.     In  analytic 

trigonometry   it   is   proved    that 

cos  6  +  i  sin  6  =  c'".*      Hence  the 

complex  variable  r{vo9,e  +  i  sin^)=  r-c'^  whore  the  stretching 

factor  (tensor)  and  turning  factor  (versor)  are  neatly  sc[)a,ratcd. 

Problems.  —  1.    Locate  the  points  rcpresmtud  by  2  +  t5;    3-i2; 

-  1  +  i2  ;  i5  ;    -  i4  ;   -  3  -  i  ;    -  t  7  ;   +  i7. 

2.  Draw   the    vectors    represented    by    3  +  i2;    1— i3;     —  2+i3; 

-  1  -i4;    -  i5  ;    3  -/;    1  + /. 

3.  Show  that  e-'"^'  =  1,  when  n  is  any  integer. 

4.  Show  that  r  •  e'(9+-"'^'  represents  the  same  point  for  all  integral 
values  of  n. 

2nni 

5.  Locate  the  different  points  represented  by  e"^  for  integral  values 
of  n. 

I(g+2n7r) 

6.  Locate  the  different  points  represented  by  5  •  e  •«  for  integral 
values  of  n. 

*This  relation  was  discovered  by  Eulcr  (1707-1783). 


162  ANALYTIC  GEOMETRY 


Akt.  83.  —  Arithmetic  Operations  applied  to  Vectors 

The  sum  of   two  complex  variables  Xj  +  ii/i  and  X2  +  iy2  is 
(.I'l  +  X,)  +  /  (^1  +  //,) .     H  euce 


|(a-i  +  iyi)  +  (•'^•2  +  ^'^2)1  =  y/{xi  +  x^y+ivi  +  2/2)', 

and.  the  amplitude  of   the  sum  is  tan"'-''      A     The  graphic 

Xj  ~|~  ^2 

representation  shows  that  the  vector  corresponding  to  the  sum 
^-rC^  +  ii/)      is  found  by  constructing  the  vec- 
'  ""      /  tor  corresponding  to  Xi  +  i?/i  ^iid. 
/  using  the  extremity  of  this  vector 
as  origin  of  a  set  of  new  axes 
w^  ^.  parallel  to  the  first  axes  to  con- 
struct   the  vector  corresponding 


^     to  X2  +  ^2/2•     The  vector  from  the 
F'G-  ^■^6.  origin  to  the  end  of  the  last  vec- 

tor is  the  vector  sum.  The  vector  sum  is  independent  of  the 
order  in  which  the  component  vectors  are  constructed.  From 
the  figure  it  is  evident  that 

I  {^1  +  Wi)  +  {^2  +  iVi)  I  >  l-^'i  +  m\  +  \^2  +  il/2\- 

The  difference  between  two  vectors  x^  +  iyi  and  Xo  +  iy^  is 
(x^  —  x.^+  i{yi  —  y^.  The  graphic  representation  shows  that 
the  vector  corresponding  to  the  difference  is  found  by  construct- 
ing the  vector  corresponding  to  x-^  +  iyi  and  adding  to  it  the 
vector  corresponding  to  —  x.2  —  iy-,-     It  is  seen  that 


I (x,  +  iy,) - {x.  +  iy^ |  =  V(a-i  -  x.y-  +  (?/i  -  y^, 

the  amplitude  of  the  difference  is  tan-^-^^' ~^^-,  and  that  the 

a-,  -  x. 

equality  of   two  complex  variables  requires  the   equality  of 

the   coefficients  of  the  real  terms  and  the  imaginary  terms 

separately. 


CO^fPLEX   V Alii  MILE 


1G: 


The  product  of  two  complex  variables  is  most  readily  found 
by  writing  these  variables  in  the  form  r^  •  e''^',  r.^  •  e'^^.  The 
product  is  i\r.2  •  e''(«»+*2',  showing  that  the  absolute  value  of  the 
product  is  the  product  of  the  absolute  values  of  the  factors  and 
the  amplitude  of  the  product  is  the  sum  of  the  amplitudes  of 
the  factors.  Hence,  writing  the  complex  variables  in  the 
form  Ti  (cos  6i  +  i  sin  ^i),  n  (cos  60  +  i  sin  O.j),  the  product  is 
'V*2[cos  (^1  +  ^2)  +  ^  sill  (^1  +  ^2)]?  which  of  course  can  be  shown 
directly. 

Construct  tlie  vector  corresponding  to  the  multiplier  r,  •  e'^* 
and  join  its  extremity  Pj  to  the  extremity  of  the  unit  vector 
01.  Construct  the  vector  corre- 
sponding to  the  midtiplicand 
7-2  •  e'^%  and  on  this  vector  OP.,  as 
a  side  homologous  to  01  construct 
a  triangle  OP^P  similar  to  OPjl ; 
then  OP  is  the  product  vec- 
tor. For,  from  the  similar  tri- 
angles 0P=  Ti  •  r^,  and  the  angle 
XOP=6i+62-  The  product  vec- 
tor is  therefore  formed  from  the 
vector  which  is  the  multiplicand 

in  the  same  manner  as  the  vector  which  is  the  multiplier  is 
formed  from  the  unit  vector.  The  product  vector  is  indepen- 
dent of  the  order  of  the  vector  factors  and  can  be  zero  only 
when  one  of  the  factors  is  zero. 

The  quotient  of  two  complex  variables  i\  •  e''^',  ?*2'  e'^-  is 

'"1 .  e'(9i  62) . 


that  is,  the  absolute  value  of  the  quotient  is  the  quotient  of  the 
absolute  values,  and  the  amplitude  of  the  quotient  is  the  ampli- 
tude of  the  dividend  minus  the  amplitude  of  the  divisor. 

Construct  the  vectors  OP,  and  OPo  corresponding  to  dividend 
and  divisor  respectively,  and  let  01  be  the  unit  vector.     On 


164 


ANALYTIC  GEOMETRY 


OPi  as  a  side  homologous  to  01\  construct  the  triangle  OI\P 
similar  to  OP^l,  then  OP  is  the  quotient  vector,  for  OP  =  -j 

and  the  angle  XOP  is  ^i  —  $2-    The  quotient  vector  is  obtained 
from  the  vector  which  is  the  dividend  in  the  same  manner  as 
the  unit  vector  is  obtained  from  the 
vector  which  is  the  divisor. 
Extracting  the  ??i  root  of 
z  =  r  •  e'^  =  r  •  e'f^+^'w) 

there  results  z"^  =  9-™ .  e  ^    "» 
Since  n  and  m  are  integers, 

Avhere  q  is  an  integer  and  r  can  have  any  value  from  0  to  m  —  l. 


<'^  +  ^ 


Hence   2;"'  =  r™-e~'"     "  ;    that   is,  the  m  root  of  z  has  m 

values  which  have  the  same  absolute  value   and   amplitudes 
differing  by  —  beginning  with  — .* 


Problems.  —  1.    Add  (2  +  1 5) ,  (  -  3  +  1 2) ,  (5  -  i  3). 

2.  Find  the  value  of  (3  -  i2)  +  (7  +  i4)  -  (0  -  i3). 

3.  Find  absolute  value  and  amplitude  of 

(4-j.3)  +  (2  +  i5)-(-3  +  i4). 

4.  Construct  (2  -  i  3)  x  (5  +  i  2)  h-  (4  -  i  5) . 

5.  Find  absolute  value  and  amplitude  of  (10  -  i  7)  x  (4  -  iS  ). 

6.  Find  absolute  value  and  amplitude  of  (15  +  f8)  x  (5  -  i2). 

7.  Construct  (2  + 1 3)3.  9.    Construct  (7  +  i  4)  ^ 

8.  Construct  (8  -  1 5)^.  10.    Construct  (9  -  i  7) t. 

*  In  mechanics  coplanar  forces,  translations,  velocities,  accelerations, 
and  the  moments  of  couples  are  vector  quantities  ;  that  is,  quantities 
which  are  completely  determined  by  direction  and  magnitude.  Hence 
the  laws  of  vector  combination  are  the  foundation  of  a  complete  graphic 
treatment  of  mechanics, 


COMPL  EX   VA  R I A  BLE 


165 


11.  Construct  the  five  fifth  roots  of  unity. 

12.  Construct  the  roots  of  2-  —  3  2  +  f)  =  0. 

Put  2  =  a;  +  ill.  There  results  (x^  _  2/2  -  3  x  +  5)  +  t  (2  xy  -  3  ?/)  =  0. 
Plot  ofl  -  y-  -  3 X  +  5  =  0  and  2xy  -  Sy  =  0.  The  values  of  z  deter- 
nihied  by  the  intersections  of  these  curves  are  the  roots  of  z'^— 3  2  +  5=0. 


AiiT.  84.  —  Algkp.raic  Functions  of  thk  Complex  Variable 

The  geonietric  representative  of  the  real  variable  is  the  point 
system  of  the  X-axis  and  the  geometric  representation  of  a 
function  of  a  real  variable  7/  =  f(x)  is  the  line  into  which  this 
function  transforms  the  X-axis. 

The  geometric  representative  of  the  complex  variable  is  the 
point  system  of  the  XF-plane,  and  the  geometric  representation 
of  a  function  of  a  complex  variable  u  +  iv=f(x-\-  i;/)  is  the 
system  of  lines  into  which  this  function  transforms  systems  of 
lines  in  the  XF-plane. 


^ 

: 

1 

' 

Y 

1 

I 

fi 

5 

^a 

A 

rf 

d' 

' 

- 

' 

' 

2 

V 

1 

: 

I 

5 

'—Ct 

\' 

A' 

d' 

Fio.  149. 

When  the  complex  variable  is  written  in  the  form  x  -f-  ??/,  it 
is  convenient  to  use  the  systems  of  parallels  to  the  X-axis  and 
to  the  F-axis.  Take  the  function  iv  =  z-\-  c,  where  iv  stands 
for  u  4-  iv,  z  for  x  -\-  vj,  c  for  a  +  ih,  then 

?t  +  iv  =  (x  -f  a)  +  /  (//  +  '>)  and  ii.  =  x-\-  a,  v  =  .y  +  h. 


166 


ANALYTIC  GEOMETRY 


If  in  the  XF-plane  a  point  moves  in  a  parallel  to  the  F-axis, 
X  is  constant,  and  consequently  u  is  constant.  Hence  the  func- 
tion 10  —  z  +  c  transforms  parallels  to  the  F-axis  into  parallels 
to  the  F-axis  in  the  f/F-plane.  In  like  manner  it  is  shown 
that  lo  —  z  +  c  transforms  parallels  to  the  X-axis  into  parallels 
to  the  {/-axis.  If  the  variables  to  and  z  are  interpreted  in  the 
same  axes,  the  function  io  =  z  +  c  gives  to  every  point  of  the 
XF-plane  a  motion  of  translation  equal  to  the  translation 
which  carries  A  to  c. 

When  the  complex  variable  is  written  in  the  form  r  •  e'^,  it  is 
convenient  to  use  a  system  of  concentric  circles  and  the  system 
of  straisfht  lines  throudi  their  common  center.     Take  the  func- 


tion lu  —  c-z,  when  to  stands  for  R  •  e'®,  z  for  r  •  e'^  and  c  for 
r'  •  e'«  ;  then  R  •  e'®  =  rr'  ■  e'(«+e  ^  and  R  =  rr',  ©  =  6  +  6'.  If  a 
point  in  the  XF-plane  describes  the  circumference  of  a  circle 
center  at  origin,  r  is  constant,  and  consequently  R  is  constant, 
and  the  corresponding  point  describes  a  circumference  in  the 
C/F-plane,  center  at  origin,  and  radius  r'  times  the  radius  of 
the  corresponding  circle  in  the  XY-plane.  If  the  point  in  the 
XF-plane  moves  in  a  straight  line  through  the  origin,  6  is  con- 
stant, and  consequently  ©  is  constant,  and  the  corresponding 


coMrLEx  V.  1  in  A  hle 


107 


p(iiiit  in  the  CF-plane  moves  in  a  straight  line  through  the 
origin.  If  the  variables  w  and  z  are  interpreted  in  the  same 
axes  X  and  Y,  the  function  xo  =  c  •  z  either  stretches  the  XY- 
plane  outward  from  the  origin,  or  shrinks  it  toward  the  origin, 
according  as  r'  is  greater  or  less  than  unity,  and  then  turns  the 
whole  plane  about  the  origin  through  the  angle  0'. 


R  = 


The  function  lo 
1    ^ 


may  be  written  R  •  e'®  : 


■'^',  whence 


z  r 

A  circle  in  the  XF-planc  with  center  at  the 


origin  is  transformed  into  a  circle  in  the  (/F-plane  with  center 
at  the  origin,  the  radius  of  one  circle  being  the  reciprocal  of 
the  radius  of  the  other.     A  straight  line  through  the  origin  in 

V 


the  XF-plane  making  an  angle  6  with  the  X-axis,  is  trans- 
formed into  a  straight  line  through  the  origin  in  the  t/F-plane 
making  an  angle  —  0  Avith  the  {7-axis.  If  iv  and  z  are  inter- 
preted in  the  same  axes,  the  function  lo  =-  is  equivalent  to  a 

z 
transformation  by  reciprocal  radii  vectors  with  respect  to  the 
unit  circle,  and  a  transformation  by  symmetry  with  respect  to 
the  axis  of  reals. 


168 


ANALYTIC  GEOMETRY 


In  the  equation   xv  =  z^,    or   R  •  e'®  =  ?-^  •  e'''^,  iv  is  a  single 
valued  function  of  z,  but  2  is  a  three-valued  function  of  w. 

Since  r  =  i2%  ^  =  ®  +  ^^^,  the  absolute  values  of  the  three 

3  3 
values  of  z  are  the  same,  but  their  amplitudes  differ  by  120°. 
The  positive  half  of  the  (7-axis,  ^  =  0,  corresponds  to  the  posi- 
tive half  of  the  X-axis,  and  the  lines  through  the  origin 
making  angles  of  120°  and  240°  with  the  X-axis.  The  entire 
C/F-plane  is  pictured  by  the  function  iv  =  z^  on  each  of  the 
three  parts  into  which  these  lines  divide  the  XF-plane. 


Art.  85.  —  Generalized  Transcendental  Functions 

Since  z  =  x^  iy  =  r  •  6'^^+""-',  log  z  =  log  r  +  i{e  +  2  mr).     The 
equation  w  =  log  2  may  be  written  u  +  iv  =  log  ?•  -f-  i  (^  -f-  2  mr). 

Y 


V 

277 

77r 
i 

37r 

55 

TT 

¥ 

y2 

A 

U 

0 

1 

f 

■'s 

Hence  u  =  log  r,  v  =  ^  +  2  htt.  To  the  circle  r  =  constant  in 
the  XF-plane  there  corresponds  in  the  C7F-plane  a  straight 
line  parallel  to  the  F-axis ;  to  the  straight  line  6  =  constant  in 
the  XF-plane  there  corresponds  in  the  f/F-plane  a  system  of 
parallels  to  the  i7-axis  at  distances  of  2  tt  from  one  anotlier ;  w  - 
is  an  infinite  valued  function  of  z,  but  2  is  a  single  valued  func- 


COMPLEX   VARIABLE 


1G9 


tion  of  IV.     Tlie  entire  Xl'-plane  is  i)ictui'ed  between  any  two 
successive  parallels  to  the  C-axis  at  distances  of  2  tt. 
Writing  the  function  ?o  =  sin  (x  +  itj)  in  the  form 
u  -\-  IV  =  sin  X  cos  iij  +  cos  x  sin  iij, 
and  remembering  that 

cosh  ?/  =  1  (e"  +  e  *)  =   cos  «//,   sinh  ?/  =  i(e^'  -  6-")=  -  i  sin  iy, 
there  results 

M  -j-  iv  =  cosh  ?/  sin  .r  —  /  sinh  ?/  cos  x, 
whence 

?t  =  cosh  ?/  sin  x,   v  =  —  sinh  ?/  cos  ^'j   and   sin  x  = 


cosh  u 


cos 


a;= ^,  cosh  ?/=-r^,    sinh^  = 

sinh  >j  sm  x  cos  x 


sma; 

Substituting  in  sin-.v+cos-a;=l  and  cosh-.y-sinli-//  =  l,  there 
results 


w 


V 


1,  — V  =  i- 

cosh-?/  '  sinh'-^y  sui^x     cos-iK 


Y 

3 

2 

1 

0 

I/,  TT 

H^ 

,,,. 

T 

X 

\V4^       \ 

V 

_3___ 
2 

//X 

1 

Uax 

V^'    /  ffA\ 

0 

irn  1       " 

Iv/  ] 

/    yyi 

— 

W- 

-'^/jU 


These  equations  when  x  and  ?/  are  respectively  constant  rep- 
resent a  system  of  confocal  conic  sections  with  the  foci  at 
(4- 1,  0),  (- 1,  0).  The  entire  system  of  ellipses  filling  up  the 
C7F-plane  is  obtained  by  assigning  to  y  values  from  +  co  to 
—  oc  ;  the  entire  system  of  hyperbolas  filling  up  the  C/F-plane 


170  ANALYTIC   GEOMETRY 

is  obtained  by  assigning  to  x  values  from  0  to  2  7r.  Hence 
v:  =  sin  (x  +  iy)  pictures  that  part  of  the  XF-plane  between 
two  parallels  to  the  F-axis  at  a  distance  of  2  tt  from  each  other 
on  the  entire  C/F-plane.* 

Problems.  —  1.    Show  that  ■?«  =  -  transforms  the  system   of  straight 
z 

lines  through  a  +  ih,  and  the  system  of  circles  concentric  at  this  point 

into  systems  of  orthogonal  circles. 

2.  Find  what  part  of  the  AT-plane  is  transformed  into  the  entire 
{/ ^''-plane  by  the  function  w  =  z"^. 

3.  Into  what  systems  of  lines  does  tp  =  cos  2;  transform  the  parallels  to 
the  X-axis  and  to  the  F-axis  ? 

*The  geometric  treatment  of  functions  of  the  complex  variable  has 
been  extensively  developed  by  Riemann  (1826-G6)  and  his  school. 


ANALYTIC    GEOMETRY    OF    THREE 
DIMENSIONS 


CHAPTER    XIII 


POINT,  LINE,  AND  PLANE  IN  SPACE 


Art. 


—  Rectilinear  Space  Coordinates 


Through  a  point  in  space  draw  any  three  straight  lines  not 
in  the  same  plane.  The  point  is  called  the  origin  of  coordi- 
nates, the  lines  the  axes  of  coordinates,  the  planes  determined 
by  the  lines  taken  two  and  two,  the  coordinate  planes.  The 
distance  of  any  point  P  from 
a  coordinate  plane  is  meas- 
ured on  a  parallel  to  that  axis 
which  does  not  lie  in  the  plane, 
and  the  direction  of  the  point 
from  the  plane  is  denoted  by 
the  algebraic  sign  prefixed  to  ^^^ 
the  number  expressing  the  dis-  ___' 
tance.  The  interpretation  of 
these  signs  is  indicated  in  the 
figure.  If  the  distance  and 
direction  of  the  point  from  the 
yZ-plane  is  given,  x  —  a,  the  ^'"-  ''"* 

point  must  lie  in  a  determinate  plane  parallel  to  the  J'Z-plane. 
If  the  distance  and  direction  of  the  point  from  the  XZ-plane  is 
given,  y  =  h,  the  point  must  lie  in  a  determinate  plane  parallel 
to  the  XZ-plane.  If  it  is  known  that  x  =  a  and  y  =  b,  the 
point  must  lie  in  each  of  two  planes  parallel,  the  one  to  the 
171 


172  ANALYTIC  GEOMETRY 

FZ-plane,  the  other  to  the  XZ-plane,  and  therefore  the  point 
must  lie  iu  a  determinate  straight  line  parallel  to  the  Z-axis. 
If  the  distance  and  direction  of  the  point  from  the  XF-plane 
2;  =  c  is  also  given,  the  point  must  lie  in  a  determinate  plane 
parallel  to  the  XF-plaue  and  in  a  determinate  line  parallel  to 
the  Z-axis ;  that  is,  the  point  is  completely  determined. 

Conversely,  to  every  point  in  space  there  corresponds  one, 
and  only  one,  set  of  values  of  the  distances  and  directions  of 
the  point  from  the  coordinate  planes.  For  through  the  given 
point  only  one  plane  can  be  passed  parallel  to  a  coordinate 
plane,  a  fact  which  determines  a  single  value  for  the  distance 
and  direction  of  the  point  from  that  coordinate  plane. 

The  point  whose  distances  and  directions  from  the  coordi- 
nate planes  are  represented  by  x,  y,  z  is  denoted  by  the  symbol 
{x,  y,  z),  and  x,  y,  z  are  called  the  rectilinear  coordinates  of  the 
point.  There  is  seen  to  be  a  "one-to-one  correspondence" 
between  the  symbol  {x,  y,  z)  and  the  points  of  space. 

Observe  that  x  =  a  interpreted  in  the  ZX-plane  represents 
a  straight  line  parallel  to  the  Z-axis ;  interpreted  in  the 
XF-plane  a  straight  line  parallel  to  the  F-axis  ;  but  when 
interpreted  in  space  it  represents  the  plane  parallel  to  the 
FZ-plane  containing  these  two  lines.  The  equations  x  =  a, 
y  =  b  interpreted  in  the  XF-plane  represents  a  point ;  inter- 
preted in  space  they  represent  a  straight  line  through  this 
point  parallel  to  the  Z-axis. 

If  the  axes  are  perpendicular  to  each  other,  the  coordinates 
are  called  rectangular,  in  all  other  cases  oblique. 

Problems.  —  1.  Write  the  equation  of  tlie  plane  parallel  to  the  rZ-plane 
cutting  the  X-axis  5  to  the  right  of  the  origin. 

2.  What  is  the  equation  of  the  FZ-plane  ? 

3.  What  is  the  locus  of  the  points  at  a  distance  7  below  the  XF-plane  ? 
Write  equation  of  locus. 

4.  Write  the  equations  of  the  line  parallel  to  the  X-axis  at  a  distance 
-f  5  from  the  XF-plane  and  at  a  distance  -  5  from  the  XZ-plane. 


POINT,   LINE,   AND  PLANE  IN   SPACE 


173 


5.  Write  the  equations  of  the  origin. 

6.  What  are  the  coordinates  of  the  point  on  the  Z-axis  10  below  the 
Xr-plane  ? 

7.  What  arc  the  equations  of  the  Z-axis  ? 

8.  What  are  the  equations  of  a  line  parallel  to  the  Z-axis  ? 

9.  Explain  the  limitations  of  the  position   of  a  point  imposed  by 
placing  X  =  -I-  5,  then  y  =  —  5,  then  z  —  -  3. 

10.  Locate  the  points  (2,  -  3,  5);  (-  2,  3,  -  5). 

11.  Locate  (0,  4,  5);  (2,  0,  -  3). 

12.  Locate  (0,  0,  -  5);  (0,  -  5,  0). 

13.  Show  that  (o,  b,  c),   {-a,  b,  c)  are  symmetrical  with  respect  to 
the  rZ-plane. 

14.  Show  that  («,  b,  c),  (—  «,  —  b,  c)  are  symmetrical  with  respect  to 
the  Z-axis. 

15.  Show  that  (a,  b,  c),  (-  a,  -  b,  -  c)  are  symmetrical  with  respect 
to  the  origin. 

Art.  87.  —  Polar  Space  Coordinates 


Let  (x,  y,  z)  be  the  rectangular  coordinates  of  any  point  F  in 
space.  Call  the  distance  from  the  origin  0  to  the  point  r, 
the  angle  made  by  OP  with  its 
projection  OP'  on  the  XF-plane 
0,  the  angle  made  by  the  projec- 
tion OP'  with  the  X-axis  <^.  r, 
(fi,  6  are  the  polar  coordinates  of 
the  point  P.     From  the  figure 

OP'  =  r  ■  cos  e, 

X  =  OP'  ■  cos  cf)  =  r  cos  6  cos  4>, 
y  =  OP'  •  sin  ^  =  r  cos  6  sin  (^, 
z  =  r  sin  6, 

formulas  which  express  the  rec-    /Y 

tangular  coordinates  of  any  point  ^"''  ^^' 

in  space  in  terms  of  the  polar  coordinates  of  the  same  point 


174 


ANAL YTIC  GEOMETll Y 


From  the  figure  are  also  obtained  r  ={x-  +  9/  +  z^)'^,  sin^  =-, 
tan  4>  = '-,  formulas  which  express  the  polar  coordinates  of  any 
point  in  space  in  terms  of  the  rectangular  coordinates  of  the 
same  point. 

Problems.  —  1.    Locate  the  points  whose  polar  coordinates  are  5,  15°, 
60"  ;  8,  90°,  45^ 

2.  Find  the  polar  coordinates  of  the  point  (3,  4,  5). 

3.  Find  the  rectangular  coordinates  of  the  point  (10,  30°,  60°). 

4.  Find  the  distance  from  the  origin  to  the  point  (4,  5,  7). 


Art. 


Distance  between  Two  Points 


Let  the  rectangular  coordinates  of  the  points  be  {x',  y\  z'), 
(x",  y",  z").     From  the  figure 

If-  =  D"  +  (z'  -  z"y,  D"  =  (x'  -  x"f  +  (?/'  -  y")\ 

hence  (1 )      D'  =  {xJ  -  x"f  +  (ij'  -  ?/")-  +  (2'  -  z"f. 

x'  —  x"  is  the  projection  of  D  on  the  X-axis;  y'  —  y"  the  pro- 


Z 


^-v., 


POINT,   LINE,   AND   PLANE  IN  SPACE  175 

jection  of  D  on  the  I'-axis;  z'  —  z"  the  projection  of  D  on  the 
^axis.*  Calling  the  angles  which  B  makes  with  the  coordi- 
nate axes  respectively  X,  Y,  Z, 

x'  —  x"  =  D  cos  X,  y'  —  y"  =  D  cos  Y,  z'  —  z"  =  D  cos  Z. 
Substituting  in  (1),  there  results 

D-  cos-  X  +  D-  cos-  Y+  D-  cos-  Z  =  D% 
whence                         cos- X  +  cos'  Y+  cos- Z=l; 
that  is,  the  sum  of   the  squares  of   the  cosines  of   the  three 
angles  which  a  straight  line  in  space  makes  with  the  rectangu- 
lar coordinate  axes  is  unity.  

The  distance  from  (x',  y',  z')  to  the  origin  is  Va;'"  +  y'^  +  z'^. 
If  the  point  {x,  y,  z)  moves  so  that  its  distance  from  {x',  y',  z') 
is  always  R,  the  locus  of  the  point  is  the  surface  of  a  sphere 
and  (x  -  x'Y  +  {y-  y'-)  +  {z-z'f  =  Er,  which  expresses  the 
geometric  law  governing  the  motion  of  the  point,  is  the  equa- 
tion of  the  sphere  whose  center  is  (x',  y\  z'),  radius  R. 

Problems.  —1.   Find  distance  of  (2,  -  3,  5)  from  origin. 

2.  Find  the  angles  which  the  line  from  (3,  4,  5)  to  the  origin  makes 
with  the  coordinate  axes. 

3.  Find  distance  between  points  ( -  2,  4,  -  5),  (3,  -  4,  5), 

4.  Write  equation  of  locus  of  points  whose  distance  from  (4,  -  1,  3) 
is  5. 

5.  Write  equation  of  sphere  center  at  origin,  (2,  1,-3)  on  surface. 

6.  The  locus  of  points  equidistant  from  (x',  y',  z'),  (x",  y",  z")  is  the 
plane  bisecting  at  right  angles  the  line  joining  these  points.  Find  the 
equation  of  the  plane. 

7.  Find  the  equation  of  the  plane  bisecting  at  right  angles  the  line 
joining  (2,  1,3),  (4,3,  -2). 

*  The  projection  of  one  straight  line  in  space  on  another  is  the  part  of 
the  second  line  included  between  planes  through  the  extremities  of  the 
first  line  peri^endicular  to  the  second.  The  projection  is  given  in  direc- 
tion and  magnitude  by  tlie  product  of  the  line  to  be  projected  into  tlie 
cosine  of  the  included  angle. 


176 


A NA L  YTIC  GEOMETR  Y 


8.    Show  that 


y'  +  ir 


+  z< 


is  the  point  midway  be- 


tween  (x',  j/',  z')^  {%",  y",  z"). 

9.   Find  the  point  midway  between  (4,  5,  7),  (2,  —  1,  3). 

10.  Find  the  equation  of  the  sphere  wliich  has  the  points  (4,  5,  8), 
(2,  —  3,  4)  at  the  extremities  of  a  diameter. 

11.  Write  the  equation  of  the  spliere  with  the  origin  on  the  surface, 
center  (5,0,  0). 

12.  Find  angles  which  the  line  through  (2,  3,  -  5),  (4,  -  2,  3)  makes 
with  the  coordinate  axes. 

13.  The  length  of  the  line  from  the  origin  to  (x,  ?/,  z)  is  ?•,  the  line 
makes  with  the  axes  the  angles  a,  /8,  7.  Show  that  x  =  rcosa,  ?/  =  rcos  3, 
z  =  r  cos  7. 


Art. 


Equations  of  Lines  in  Space 


Suppose  any  line  in  space  to  be  given.  From  every  point  of 
the  line  draw  a  straiglit  line  perpendicular  to  the  XZ-plane. 
There  is  formed  the  surface  which  projects  the  line  in  space 
on  the  XZ-plane.  The  values  of  x  and  z  are  the  same  for  all 
points  in  the  straight  line  which  projects  a  point  of  the  line 

in  space  on  the  XZ-plane. 
Hence  the  equation  of  the 
projection  of  the  line  in 
space  on  the  XZ-plane  when 
interpreted  in  space  repre- 
sents the  projecting  surface. 
The  projection  of  the  line  in 
space  on  the  XZ-plane  deter- 
mines one  surface  on  which 
/Y  the   line    in   space    must  lie. 

The  projection  of  the  line 
in  space  on  the  I''Z-plane  determines  a  second  surface  on 
which  the  line  in  space  must  lie.  The  equations  of  the  pro- 
jections of  the  line  in  space  on  the  coordinate  planes  XZ  and 


POLXT,    LINE,   AND   PLANE  IN  SPACE 


177 


YZ  therefore  determine  the  line  in  space  and  are  called  the 
equations  of  the  line  in  space.  By  eliminating  z  from  tlie 
equations  of  the  projections  of  the  line  on  the  planes  XZ  and 
YZ,  the  equation  of  the  projection  of  the  line  on  the  XF- 
plane  is  found. 


Art.  90.  —  E(juATroNs  of  the  STUAKiiix  Link 


iglit  line  on  the 
h  «,  y  =  bz  +  (3. 


The  equations  of  the  projections  of  the  sti 
coordinate    planes    XZ  and    YZ    are    x  =  az 
The  geometric  meaning  of 
a,  b,  a,  fi  is  indicated  in 
the  figure.     The   elimina- 
tion of  z  gives 

y-(3  =  l{x-a), 

the  equation  of  the  pro- 
jection of  the  line  in  the 
XF-plane. 

Two    points, 

(x',  y',  z'),  {x",  y",  z"), 
determine  a  straight  line 
m  space.  The  projection 
of  the  line  through  the 
points  {x\  y,'  z'),  (x",  y",  z")  on  the  ZX-plane  is  determined  by 
the  projections  (a;',  z'),  (x",  z")  of  the  points  on  the  ZX-plane, 
likewise  the  projection  of  the  line  on  the  ZF-plane  is  deter- 
mined by  the  points  {z',  y'),  (z",  y").  Hence  the  equations  of 
the  straight  lines  through  {x',  y',  z'),  (x",  y",  z")  are 


Fm.  159. 


^-(z-z'),  y-y'  = 


-(z-z'). 


A  straight  line  is  also  determined  by  one  point  and  the  direc- 
tion of  the  line.     Let  {x',  y',  z')  be  one  point  of  the  line,  «,  /3,  y 

N 


178 


ANALYTIC   GEOMETRY 


the  angles  which  the  line  makes  with  the  axes  X,  Y,  Z  respec- 
tively. Let  {x,  y,  z)  be  any  point 
of  the  line,  d  its  distance  from 
{x',  y',  z').     Then 

z-x'  ^y-y'  ^z-z'  ^_^ 
cos  a  cos  13  cos  y 
is  the  equation  of  the  line.  This 
equation  is  equivalent  to  the  equa- 
tions x=x'  +  d  cos  a,  y=y'+d  cos  (3, 
z  =  z'  +  d  cos  y,  which  express  the 
coordinates  of  any  point  of  the  line 
in  terms  of  the  single  variable  d. 

If  the  straight  line  (1)  contains 
the  point  (x",  y",  z"), 
x"-x'  ^  y"-y'  ^  z"  -z'  , 

cos  a         cos  /8        cos  y 
Eliminate  cos  a,  cos  (3,  cos  y  from  (1)  and  (2)  by  division,  and 
the  equation  of  the  straight  line  through  two  points  is  obtained 
x  —  x'  _  y  —  y'  _  z  —  z' 
x"  —  x'     y"  —  y'      z"  —  z' 
as  found  before,     a,  (3,  y  are  called  the  direction  angles  of  the 
straight  line. 

Problems.  —  1.   The  projections  of  a  straight  line  on  the  planes  XZ 
y     z 


Find  the  projection  on  the  XY 
-  5,  ?/  =  2  2  —  3  with  the  coordinate 


and   YZ  are  2  x  +  3 
plane. 

2.  Find  the  intersections  of  x 
planes. 

3.  Write  the  equations  of  the  straight  line  through  (2,  3, 1),  ( -  1,  3,  5) . 

4.  Write  the  equations  of  the  straight  line  through  the  origin  and  the 
point  (4,  -  1,  2). 

5.  Write  the  equations  of  the  straight  line  through  (3,  1,  2)  whose 
direction  angles  are  (60°,  45°,  G0°). 

6.  The  direction  angles  of  a  straight  line  are  (45°,  60°,  60°) ;  (4,  5,  6) 
is  a  point  of  the  line.     Find  the  coordinates  of  the  point  10  from  (4,  5,  6). 


POINT,   LINE,    AND   PLANE  IN    SPACE 


ITU 


Akt.  91. 


Let 


X  —  a 
cos  « 


Angle  hetwkex  Two  Stuaigux  Lines 

X 


y-b  ^z~c 
cos  (i      cos  y 


if 


cos  /8' 


cos  a'  cos  li'  cos  y 
be  the  straight  lines.  The  angle  between  the  lines  is  by  definition 
the  angle  between  parallels  to 
the  lines  through  the  origin. 
Let  OM'  and  OM"  be  these 
parallels  through  the  origin. 
From  any  point  P'{x\  y\  z') 
of  OM'  draw  a  perpendicular 
P'P"  to  OM".  Then  OP"  is 
the  projection  of  OP'  on  OM", 
and  OP"  is  also  the  projection 
of  the  broken  line  {x'  +  y'  +  z') 
on  OM".*     Hence  / 

r'  cos  6  =  x'  cos  a'  +  y'  cos  ^' 

+  Z'  cos  y, 


=  '-cos  a  -|-— cos 
r'  y' 


+  -C0Sy 
7 


that 


cos  0  =  cos  a  COS  a'  +  cos  /3  cos  (3'  +  cos  y  cos  y'. 


(1) 


*  The  sum  of  the  projections  of 
the  parts  of  a  broken  line  on  any 
straight  Hne  is  the  part  of  the  line 
included  between  the  projections  of 
tlie  extremities  of  the  broken  line. 
a!)  is  the  projection  of  AB  ;  be  is  the 
projection  of  BC;  ac  is  the  projec- 
tion of  AB  +  BC. 


Fio.  162. 


180  ANALYTIC   GEOMETRY 

If  the  equations  of  the  lines  are  written  in  the  form 

X  =  az  +  a,  y  —  bz  +  (3]  x=  a'z  +  «',?/  =  b'z  +  (3', 

the  equations  of  parallel  lines  through  the  origin  are 

X  =  az,  y  =  bz ;  x  =  a'z,  y  —  b'z. 
Let  (x',  y',  z')  be  any  point  of  the  first  line,  its  distance  from 
the  origin  r'.    Then  x'  —  az',  y'  —  bz',  r'?  =  x'-  +  y'^  +  z'-,  whence 


cos  a 

x' 
~  r' 

r' 

= 

a 

VI 

cos/8 
cosy 

b 

+ 

b' 

vr 

+  a- 
1 

T 

1?' 

Likewise  if  {x",  y",  z")  is  any  point  of  the  second  line,  r"  its 
distance  from  the  origin. 


Substituting  in  (1) 


cos  ii' 

x" 
r" 

a' 

V'l  -f-  «'- 

■■  +  b'^ 

cosfi' 

r" 

b' 

VI  +  a"^ 

'■  +  b" 

cosy' 

_z" 
r" 

1 

VI  +  a" 

+  b" 

l+aa'  +  bb' 

Vl  +  a^  +  bWl  +  a"  +  b" 
When  the  lines  are  perpendicular,  cos  ^  =  0,  whence 

1  +  aa'  +  bb'  =  0. 
When  the  lines  are  parallel,  cos^  =  1,  whence 
^  _  1  +  aa,'  +  bb' 


Vl  +  a-  +  bWl  +  a"  +  b" 
which  reduces  to 

(a'  -  ay  +  (b'  -  by  +  (ab'  -  a'by  =  0. 


POINT,    LINE,   AND   PLANE   IN    SPACE  181 

This  eciuation  requires  that  a  =  a',  b  =  b' ;  that  is,  if  two  lines 
are  parallel,  their  projections  on  the  coordinate  planes  are 
parallel. 

The  equations  of  the  straight  line  through  (x',  ?/',  z')  parallel 
to  x=az  + a,  y==bz-{-  ^  are  a;  —  x'=  a {z  —  z'),  y  ~  y'  =  b{z  —  z'). 

The  straight  line  (1)  x  —  x'  =  a'(z  —  z'),  y  —  y' =  b' {z  —  z') 
through  the  point  (x',  y',  z')  is  perpendicular  to  the  straight  line 
(2)  X  =  az  +  a,  y  =  bz  +  ft  when  a'  and  b'  satisfy  the  equation 
1  +  cm'  +  bb'  =  0.  This  equation  is  satisfied  by  an  infinite 
number  of  pairs  of  values  of  a'  and  b'.  This  is  as  it  ought  to 
be,  for  through  the  given  point  a  plane  can  be  passed  perpen- 
dicular to  the  given  line,  and  every  line  in  this  plane  is  perpen- 
dicular to  the  given  line,  and  conversely.  Hence  if  the  straight 
line  (1)  is  governed  in  its  motion  by  the  equation  l  +  aa'-j-6^'  =  0, 
it  generates  the  plane  through  (x',  y',  z')  perpendicular  to  the 
straight  line  (2).  1  -f  aa'  -\-  bb'  =  0  is  the  line  equation  of  the 
plane. 

To  find  the  relation  between  the  constants  in  the  equa- 
tions of  two  straight  lines  x  =  az  +  a,  y  =  bz  +  ^,  x  =  a'z  +  a', 
y  —  b'z  +  /3',  which  causes  the  lines  to  intersect,  make  these 
equations  simultaneous  and  solve  the  equations  of  the  projec- 
tions on  the  XZ-plane,  also  the  equations  of  the  projections  on 

the  I'Z-plane,  for  z.     The  two  values  of  z,    and  ^^ ^ 

a  —  a'  b  —  b' 

must  be  equal  if  the  lines  intersect.  Hence  for  intersection 
the  equation  (a  —  a')  (y8'  —  ft)  —  (b  —  b')  {a'  ~  a)  =  0  must  be 
satisfied,  and  the  coordinates  of  the  point  of  intersection  are 

aa'  —  a'a  bB'  —  b'B  «'  —  a        -.^r,  ,         -, 

x  = ,   y  = -^ -,    z  = When    a  =  a     and 

a  —  a'  b  —  b'  a  —  a' 

I)  =  //,  the  point  of  intersection  is  at  infinity,  and  the  lines  are 
parallel,  as  found  before. 

Problems.  —  1.    Find  the  angle  between  the  lines 

x  =  Sz  +  \,  y  =  ~  22  +  5;  x  =  z  +  2,  y  =  -  z  +  i. 
2.    Find  the  angle  between  the  lines  through  (1,  1,  2),  (-3,  -  2,  4) 
and  (2,  1,  -  2),  (3,  2,  1). 


182 


ANALYTIC  GEOMETRY 


3.  Find  equations  of  line  through  (4,  -  2,  3)  parallel  to  a:  =  4  2:  +  1, 
y  =  2  z  —  b. 

4.  Find  line  through  (1,  -  2,  3)  intersecting  x  =  -2z-\-o,  y=z  +  5 
at  right  angles. 

5.  Find  distance  from  (2,  2,  2)  to  line  x  =  2  z  -\-  l,  y  =  -  2  z  +  S. 

6.  Find  equations  of  line  intersecting  each  of  the  lines  x  =  3^  +  4, 
y  =  -z  +  2  and  y  =  2  z  -  5,  x  -  -  z  +  2  at  right  angles. 

7.  For  what  value  of  a  do  the  lines   x  =  Sz  +  a,    y  =  2z  +  5   and 
X  =  -  2  z  -  o,  y  =  i  z  -  d  intersect  ? 

8.  Find  the  equations  of  the  straight  line  through  the  origin  intersect- 
ing at  right  angles  the  line  through  (4,  2,  -  1),  (1,  2,  -  3). 

9.  Find    distance    of    point   of    intersection   of    lines     x  =  2z-\-l, 
y  =  2z  +  2  and  x  =  z  +  5,  y-iz-6  from  origin. 

10.    Find  distance  from  origin  to  line  x-iz-H,  y  =  —  2z  +  3. 


Art. 


The  Plane 


A  plane  is  determined  when 
the  length  and  direction  of  the 
perpendicular  from  the  origin 
to  the  plane  are  given.  Call 
the  length  of  the  perpendicular 
p,  the  direction  angles  of  the  per- 
pendicular a,  (3,  y.  Let  P(.i-,  ?/,  z) 
be  any  point  in  the  plane.  The 
projection  of  the  broken  line 
(;f  +  y  +  z)  on  the  perpendicular 
OP'  equals  p  for  all  points  in  the 
plane  and  for  no  others.  Hence 
xcosa  +  y cos  /8  +  2:  cos y  =  p  is 
the  equation  of  the  plane.  This 
is  called  the  normal  equation  of 
the  plane. 
Every  first  degree  equation  in  three  variables  when  inter- 
preted  in  rectangular  coordinates   represents  a  plane.      The 


POINT,    LINE,   AND   PLANE   LW    SPACE  183 

locus   reprostMiiod   by  Ax  +  B>i  +  Cz  +  D  =  0  is  the  same  as 
the  locus  represeuted  by  u;cos  a  +  ij  (ios  fi  +  z  cos  y  -  2'  =  ^  i^' 


Con 


cos  a 
A 

cos^ 
B 

_  cos  y 

c 

D 

ibiniiu 

;  witli 

c  +  cos- 13  +  cos 

r-y=\ 

,  cos  a 

A 

V^i- 

-\-B'+G' 

cos 

3                 B 

cos  y  = 

C 

"^      V^-  +B'  +  C 

V^l'  +  B'  +  C 

P  =  - 

D 

V^l- 

+  B'  +  C 

ice  the 

factor 

] 

_ 

Va:'  +  B-+  c- 

transfonns  Ax  +  Bi/  +  Cz  +  D  =  0  into  an  e(iuatiou  of  the  form 
.1-  cos  a  +  y  cos  (3  +  z  cos  y  =  p,  which  is  the  equation  of  a 
plane. 

•^  ^-  •'/-)-?=:  1   is  the  equation  of  the  plane  whose  intercepts 

a      b      c 
on  the  coordinate  axes   are   a,  b,  c.      This   is   the   intercept 
equation  of  the  plane. 

The  plane  represented  by  the  equation  Ax  +  By  +  Cz-{-  D  —  0 
depends  on  the  relative  values  of  the  coefficients.  Hence  the 
equation  of  the  plane  has  three  parameters.  To  find  the  equa- 
tion of  the  plane  through  three  points  {x\  y',  z'),  (_x",  y",  z"), 
{x'",  y'",  z'"),  substitute  these  coordinates  for  x,  y,  z  in 
(1)  A'x  +  By  +  C'a;  +  1  =  0,  solve  the  resulting  e(iuations  for 
A',  B',  C,  and  substitute  in  (1). 

The  intersections  of  a  plane  with  the  coordinate  planes  are 
called  the  traces  of  the  plane  on  the  coordinate  planes.  The 
equation  of  the  trace  of  Ax  +  By +Cz  +  D  =  0  on  the  X^-plane 
is  found  by  making  ?/  =  0  in  the  equation  of  the  plane.  The 
trace  is  therefore  Ax  +  Cz  +  D  =  0.  The  trace  on  YZ  is 
By  +  Cz  +  D  =  0,  on  XY is  Ax  +  By  +  D  =  0. 


184  ANALYTIC  GEOMETRY 

For  points  in  the  intersection  of  the  planes 
Ax  +  By+Cz  +  D  =  0  and  A'x  +  B'y  +  C'z-\-D'  =  0 
these  equations  are  simultaneons.     Eliminating  >j, 

{AB'  -  A'B)x  +  (CB'  -  C'B)z  +  {DB  -  D' D)  =  0, 
the  equation  of  the  projection  of  the  intersection  on  the  co- 
ordinate plane  XZ.     In  like  manner  the  equations  of  the  pro- 
jections of   the  intersection  on  the  planes  YZ  and  XF  are 
obtained. 

Problems.  —1.  Write  the  equation  of  the  plane  whose  intercepts  on  the 
axes  are  2,  —  4,  —  3. 

2.  Find  the  equation  of  the  plane  through  (2,  -  3,  4)  perpendicular 
to  the  line  joining  this  point  to  the  origin. 

3.  Find  the  equation  of  the  plane  through  (2,  5,  1),  (3,  2,   -5), 
(1,  -3,7). 

4.  Find  the  equations  of  the  traces  of3a;-?/  +  5z—  15  =  0. 

5.  Find  the  equations  of  the  intersection  of  Sx  +  by  ~  7  z  +  10  =  0, 
bx  -Utj  +  Sz  -  lb  =  0. 

6.  Find  the  equation  of  the  plane  through  (3,  -  2,  5)  perpendicular  to 
a:  -1   _  ?/  +  2  _  g  -  3 

cos  GO'^      cos  45"^      cos  G0° 

n.   Find  the  direction  angles  of  a  perpendicular  to  the  plane 

2x-3?/+52  =  6. 
8.   Find  the  length  of  the  perpendicular  from  the  origin  to 

2x-3y  +  bz=^G. 


Art.  93.  —  Distance  from  a  Point  to  a  Plane 

Let  (x',  y\  z')  be  a  given  point,  cc  cos  «  +  2/  cos  ^  +  2  cos  y  =i>, 
a  given  plane.     Through  {x\  y',  z')  pass  a  plane  parallel  to  the 
given  plane.     The  equation  of  this  parallel  plane  is 
a;  cos  «  -h  2/  cos  ^  +  »  cos  y  =  OF". 


POINT,   LINE,   AND   PLANE  IN  SPACE 


m 


The  point  (x',  y',  z')  lies  in  this  plane,  therefore 
x'  cos  a  +  y'  cos  ^  +  2'  cos  y  =  OP". 
Subtracting  OP  from  both 
sides  of  this  equation, 
x'  cos  a  +  y'  cos  fi 

-f-  z'  cos  y  —  i^  =  P-P" ; 
that  is,  the   perpendicular 
distance  from  (x',  y',  z')  to 
xcosa  +  y  cos  (3 

+  2  cos  y  —  p  =  0 
is  the  left-hand  member  of 
this  equation  evaluated  for 
(x',  y',  z').  The  sign  of  the 
perpendicular  is  plus  when 
the  origin  and  the  point 
(x',  y\  z')  are  on  different 
sides  of  the  plane,  minus 
when  the   origin   and   the 

,        /,,,■.  jl  1'  1(1.     K«. 

point  (a; ,  y',  z)  are  on  the 
same  side  of  the  plane. 

The  distance  from  {x',  y',  z')  to  the  plane  Ax-\-By+Cz-\-D=0 
is  found  by  transforming  the  equation  of  the  plane  into  the 
form  X  cos  a  -{-  y  cos  13  +  z  cos  y  —  j?  =  0  to  be 
Ax'  +  By'  +  Cz'  +  D 

Let  .Tcos  a-\-y  cos  /?  +  2:  cos  y  —  p  =  0  and 

X  cos  «'  -f  ?/  cos  )8'  -f  2  cos  y'  —  ;y  =  0 
bo  the  faces  of  a  diedral  angle, 
(.«  cos  «+?/ cos  ^+2;cos  y— ^))±(.);  cos  «'+?/ cos  ^'  +  z  cosy'— p')=0 

is  the  equation  of  the  locus  of  points  equidistant  from  the 
faces:  that  is,  the  eipiation  oC  the  bisectors  of  the  diedral 
angle. 


186  ANALYTIC  GEOMETRY 

Problems.  —  1.    Find  distance  from  origin  to  plane 
Ux-13y+nz  +  22  =  0. 

2.  Find  distance  from  (3,  -  2,  7)  to  3  a;  +  7  ?/  -  10  s  +  5  =  0. 

3.  Write  the  equations  of  the  bisectors  of  the  diedral  angles  whose 
faces  are  2  X  +  5  y  —  7  z  =  10,  and  ix-y  +  Gz  —  l!i  =  0. 

4.  Find  distance  from  (0,  5,  7)  to  -  +  |  +  ?  =  1. 

5.  Find  distance  from  origin  to  |  ce  —  |  ?/  -|  |  ^  =  1- 

Art  94.  —  Angle  between  Two  Planes 

Let  a;  cos  a  -\-  y  cos  ^+z  cos  y=2^,  a;  cos  «'+?/cos^'+2:cosy'=jy 
be  two  given  planes,  0  their  included  angle.  The  angle  be- 
tween the  planes  is  the  angle  between  the  perpendiculars  to 
the  planes  from  the  origin.     Hence 

cos  6  =  cos  a  cos  a'  +  cos  (3  cos  ^'  -\-  cos  y  cos  y'. 
If  the  equations  of  the  planes  are  in  the  form 

Ax  +  B>j  +  Cz  +  D  =  0,    A'x  +  B'y  +  C'z  +  Z)'  =  0, 


cos  a  = ■ — > 

V.4^  +  B'+  C 

COS  «■  = ? 

cosB  —             ^            > 

cos  IS'  -             ^'            , 

V^-  +  B'+  C 

^AJ'  +  B'-'+C" 

0 

pi 

cos  y'  - 

V^»  +  B'  +  C- 

V^'-  +  B"  +  C" 

r.no.     .„...-             AA'  +  BB'+CC            . 

VA'  +  B'+  CWA'^  +  B''  +  C"2 

The  planes  are  perpendicular  when  AA'  +  BB'  +  CC  =  0 ; 

,,  ,       ,         .                  AA'  +  BB'  +  CC  1  .  -, 

parallel    when    1  =  —  -'— — — ' r — =r — ,      which 

V2-  +  B'  +  C-VA"  +  B'-  +  C- 
reduces    to    {AB'  -  yl'i^)^  +  {AC  -  A'Cf  +  (7JC"  -  B'Cf  =  0, 
1  A       B      C 


POINT,    LINE,    AND    PLANE   IN   SPACE  1ST 

The  angle  between  the   plane  xcos a -\->/ con  f3  +  zcosy  =p 
y  ~  •'  =z    ~ '^    is  the  conipUnnent  of  the 


cos «'  cos  /8'  cos  y' 
angle  between  the  line  and  the  perpendiculai-  to  the  plane. 
Hence  sin  6  =  cos  a  cos  a'  +  cos  ft  cos  /3'  +  cos  y  cos  y'.  If  the 
equations   of    line    and    plane    are    in   the    form    x  =  az  + «, 

y  =  hz  +  p,  and  Ax  +  By -\- Cz-{- D  =  0, 


cos  a  = 

Vyl-  +  B'-^  C 

cos/8  = 

B 

V^l-  +  B'  +  C" 

COSv  = 

C 

a 


coS|8' 


COS  y' 


Vl  +  a2  +  62 

Vl  +  a-  4-  b^' 
1 


V^-  +  i3-  +  C  -  '  V 1  +  a-  +  62 

Hence  sin^  = ^a  +  m+C  _. 

VJ^M^-  +  C- Vl  +  a-  +  b^ 

The  line  is  parallel  to  the  plane  when  ^hi  -\-  Bb  -j-  C=0; 
perpendicular   when    1=  Aa  +  Bb  +  C  ^^^^^^^^ 

VA'  +  B'+  (J-  Vl  +  a'  +  b^ 
reduces    to    {Ab  -  Baf  -\-  {A  -  Ccif  +  (B  -  Cbf  =  0,    whence 

.1    ,      B 

a  =  —,  b  =  — 

(f         C 

To  find  the  intersection  of  the  line  x  =  az  -{-  a,  y  —  bz  -j-  (S, 
and  the  plane  Ax  +  By  +  Cz  +  J9  =  0,  make  these  equations 
simultaneous,  and  solve  for  x,  y,  z.     There  results 

^  _     Au  +  B^  +  D 

Aa  +  Bb  +  C 

If  Aa  -j-  Bb  +  C—0,  the  point  of  intersection  goes  to  infinity, 
and  the  line  and  plane  are  parallel,  as  found  before.  If 
Aa  -I-  B(3  +  C  also  vanishes,  z  lieconies  indeterminate,  likewise 
x  and  //,  and  the  line  lies  wholly  in  the  plane. 


188  ANALYTIC  GEOMETRY 

If  the  plane  Ax-\- By  +  Cz-\-  D  =  0  contains  the  point 
(x',  y',  z')  and  the  line  x  =  az  +  a,  y  =  bz-{-^, 

Ax'  +  By'  +  Cz'+D  =  0,   Aa  -\- Bb  +  C=  0,  Aa  +  B(3-\-D  =  0. 

These  equations  determine  the  relative  values  of  A,  B,  C,  D, 
hence  the  plane  is  determined. 

The  plane  Ax  +  By  -{-  Cz  +  D  =  0  contains  the  two  lines 
X  =  az  +  a,  y  =  bz  +  (^  and  x  =  a'z  +  a',  y  =  b'z  +  (3'  when 
Aa  +  Bb-^C^O,  Aa  +  Bf3  +  D^0,  Aa'  +  Bb'  +  C^O, 
Aa'  +  B/3'  +  D  =0.     These  four  equations  are  consistent  only 

when —  —  '^^'^      that  is,   when  the  lines  intersect,  and 

b—b'      /?  — /3 

then  the  relative  values  of  A,  B,  C,  D,  which  determine  the 

plane,  are  found  by  solving  any  three  of  the  four  equations. 

Problems.  —  1.  Find  angle  between  planes  10  x  —  3  y  +  4  £•  +  12  =  0, 
15  X +  11?/ -7  2 +  20  =  0. 

2.  Find    angle  between  line    x  =  5  z  +  7,    y  =  S  z  —  2,    and    plane 
2x- 15?/ +  200  + 18=0. 

3.  Find    equation    of     plane     through     (4,    —  2,    3)     parallel    to 
3x-2y  +  z-  5  =  0. 

4.  Find  equation  of  plane  through  (1,  2,  —  1)  containing  the  line 
x  =  2z  —  S,   y  =  z  +  a. 

5.  Find  equation  of  line  through  (4,  2,  —  3)  perpendicular  to 
x  +  Sy  -2z  +  4  =  0. 

6.  Find  equation  of  plane  containing  the  lines  x  =  2z  +  \,  y  =  2z  +  2, 
and  x  =  z  +  5,  y  =  4:Z  —  G. 

7.  Find  angles  which  Ax  +  By  +  Cz  +  D  =  0  makes  with  the  coordi- 
nate axes. 

8.  Find  angles  which  Ax  +  By  +  Cz  +  D  ■=  0  makes  with  the  coordi- 
nate planes. 

9.  Show  that  if  two  planes  are  parallel,  their  traces  are  parallel. 

10.  Show  that  if  a  line  is  perpendicular  to  a  plane,  the  projections  of 
the  line  are  perpendicular  to  the  traces  of  the  plane. 


POINT,   LINE,   AND   PLANE  IN   SPACE  189 

11.  Show  that  ^  ~  ^'  =  ^''s:Jl!.  =  ?-IiA  is  perpendicular  to 

A  B  C 

Ax  +  ny  +  a::  +  D  =  0. 

12.  Show  that  (x>-x"){x-x")  +  (7j' -y")(>/~y")  +  (z' -z")(:-z")=0 
is  a  plane  through  (x",  y",  z")  perpendicular  to  the  line  through 
(x',  y\  z')  and  (x",  y",  z"). 

13.  Find  the  equation  of  the  plane  tangent  to  the  sphere  x^  +  y'^+z'^  —  R- 
at  the  point  (x",  y",  z")  of  the  surface. 

14.  Find  the  equation  of  the  plane  tangent  to  the  sphere 

(X  -  X')-  +  (2/  -  y'y  +  {s-  z'Y  =  R- 
at  the  point  (x",  y" ,  s")  of  the  surface. 


CHAPTER    XIV 


OUKVED  SURPAOES 


trix   of    a   cylindrical    surface 
Z 


Art.  95.  —  Cylindrical  Surface.s 

Let  the  straight  line  x  =  az  +  a,  y  =  bz  -\-  (3  move  in  such  a 
manner  that  it  always  intersects  the  XF-plane  in  the  curve 
F(x,  y)  =  0,  and  remains  parallel  to  its  first  position.  The 
straight  line  is  the  generatrix,  the  curve  F{x,  y)=0  the  direc- 

The  generatrix  pierces  the 
XF-plane  in  the  point  («,  (3), 
and  therefore  F(a,  (3)^0. 
This  is  the  line  equation  of 
the  cylindrical  surface,  for 
since  a  and  b  are  constant,  to 
every  pair  of  values  of  a  and 
13  there  corresponds  one  posi- 
tion of  the  generatrix,  and 
to  all  pairs  of  values  of  a 
and  13  satisfying  the  equation 
F(a,  13)  =0  there  corresponds 
the  generatrix  in  all  positions 
*''*"■  '^^^-  while  generating  the  cylindri- 

cal surface.  To  obtain  the  equation  of  the  cylindrical  sur- 
face in  terras  of  the  coordinates  of  any  point  (.r,  y,  z)  of  the 
surface,  substitute  in  F{(z,  /8)=0  the  values  of  a  and  ^  ob- 
tained from  the  equations  of  the  generatrix.  There  results 
F(:x-az,  y-bz)=0,  the  equation  of  the  cylindrical  surface 
whose  directrix  is  F(x,  y)  =  0,  generatrix  x=az-\-a,  y=bz  +  (3. 
190 


CURVED   SURFACES 


11)1 


+  ^  =  1.     What  does  this  equation  become  when  elements  are  parallel 
6- 


ele- 


Problems.  —  1.  Find  the  equation  of  the  right  circular  cylinder  whose 
directrix  is  x-  +  y~  —  f-,  and  axis  the  Z-axis. 

2.  TIic  directrix  of  a  cylinder  is  a  circle  in  the  A'l'-plane,  center  at 
origin.  The  element  of  tlie  cylinder  in  the  ZA'-plane  makes  an  angle  of 
45°  with  the  A'-axis.     Find  equation  of  surface  of  cylinder. 

3.  Find  general  equation  of   surface  of  cylinder  whose  directrix  is 

X-  ,  y^ 

to  Z-axis  ? 

4.  Find  equation  of  cylindrical  surface  directrix  y-  —  \(ix 
ments  parallel  to  x  =  22  +  5,  ?/=  —  3^  +  5. 

5.  Determine  locus  represented  by 

a;  =  a  sin  </>,  y  =  a  cos  0,  z  =  r(p. 

Since  x-  +  y-  =  a-,  the  locus  must  lie  on 
the  cylindrical  surface  whose  axis  is  the 
Z-axis,  radius  of  base  a.  Points  corre- 
sponding to  values  of  (p  differing  by  2  tt  lie 
in  tlie  same  element  of  the  cylindrical  sur- 
face. The  distance  between  the  successive 
points  of  intersection  of  an  element  of  the 
cylindrical  surface  with  the  locus  is  2  wc. 
The  locus  is  tlierefore  the  thread  of  a  cylin- 
drical screw  with  distance  between  threads 
2  TTC.    The  curve  is  called  the  helix. 


Art.  96.  —  Conical  Surfaces 


Let  the  straii^^ht  lino  x  =  az  -f  «,  y  =  hz  +  (i  move  in  such  a 
manner  that  it  always  intersects  the  XT-plane  in  the  curve 
F(x,  ?/)  =  0,  and  passes  through  the  point  {x\  ?/',  z').  The 
straight  line  generates  a  conical  surface  whose  vertex  is 
(:r',  ?/',  z'),  directrix  F{x,  y)  =  0.  The  equations  of  the  generatrix 
are  x  —  x'  =  a(z  —  z'),  y  —  ?/'  =  h{z  —  z'),  which  may  be  written 
x=az-{-{x'—az'),  y=bz-{-(y'—hz').     This  line  pierces  the  XY- 


192 


Analytic  geometry 


plane  in  {x'  —  az\  y'  —  bz'),  and  therefore  F(x'—az',  y'  —  bz')=0. 
This  is  the  line  equation  of  the 
conical  surface,  for  to  every  pair 
of  values  of  a  and  b  there  corre- 
sponds one  position  of  the  gen- 
eratrix, and  to  all  pairs  of  values 
of  a  and  b  satisfying  the  equa- 
tion F(x'  —  az',  y'—bz)  =  0  there 
corresponds  the  generatrix  in  all 
positions  while  generating  the 
conical  surface.  To  obtain  the 
equation  of  the  conical  surface 
in  terms  of  the  coordinates  of 
any  point  (x,  y,  z)  of  the  surface, 
substitute  in  F(x'  —  az',  y'  —  bz')=  0  for  a  and  b  their  values 
obtained  from  the  equations  of  the  generatrix.     There  results 

j^rx'z  -  xz'   y'z-yz'\^  ^^^  equation  of  the  conical  surface 

\  z  —  z'        z~z'   J 
whose  vertex  is  (x',  y',  z'),  directrix  F(x,  y)  =  0* 


Problems.  —  1.  Find  the  equation  of  the  surface  of  the  right  circular 
cone  whose  axis  coincides  with  the  Z-axis,  vertex  at  a  distance  c  from  tlie 
origin. 

2.  Find  the  equation  of  the  conical  surface  directrix  ^  +  l^=  1,  ver- 
tex (5,  2,  1). 

3.  Find  the  equation  of  the  conical  surface  vertex  (0,  0,  10),  directrix 
2/2  =  10  X  -  x~. 

4.  Find  the  equation  of  the  conical  surface  vertex  (0,  0,  c),  directrix 
^  +  ?^'=1. 

5.  Find  the  equation  of  the  conical  surface  vertex  (0,  0,  10),  directrix 
a;'^  +  2/2^0. 


*  Surfaces  which  may  be  generated  by  a  straight  line  are  called  ruled 
surfaces. 


CURVED   sun FACES 


19a 


Art.  97.  —  Sukkacks  ok  Kkvolution 


Let  3fN  be  any  line  in  the  ZX-plane.  When  MN  revolves 
about  the  Z-axis,  every  point  /*  of  JfiV  deseribes  the  circumfer- 
ence of  a  (drcle  witli  its  center 
on  the  Z-axis  and  whicli  is  pro- 
jected on  the  XF-plane  in  an 
equal  circle.  The  equation  of 
the  circle  referred  to  a  pair  of  (' 
axes  through  its  center  parallel 
to  the  axes  X  and  Y  is 
or  -j-y-  =  t^. 
This  is  also  the  equation  of  the 
})rojectiou  of  the  circle  on  the 
Xl''-})lane.  The  radius  r  is  a 
function  of  z  whicli  is  given  by 

the  equation  of  the  generatrix  r  =  F(z).  Hence  the  equation 
of  the  surface  of  revolution  is  obtained  by  eliminating  /•  from 
the  equations  af'  +  y-  —  r'  and  r  =  F(z). 


Problems.  —  1.  Find  equation  of  surface  of  sphere,  center  at  origin, 
radius  li.  This  sphere  is  generated  by  the  revolution  about  the  Z-axis  of 
a  circle  whoso  e(iualion  is  r~  +  z-  =  R-.  Eliminate  r  from  this  equation 
and  x-  -|-  y-  —  ?•-,  and  the  ecpiation  of  the  sphere  is  found  to  be 

X-'  -f  if  -V  Z'^:=  R\ 

2.  Find  equation  of  right  circular  cylinder  vyliose  axis  is  the  Z-axis. 

3.  Find  equation  of  right  circular  cone  whose  axis  is  Z-axis,  vertex 
(0,  0,  c). 

4.  Find  equation  of  right  circular  cone  whose  axis  is  Z-axis,  vertex 
(0,  0,  0). 

5.  Find  equation  of  surface  generated  by  revolution  of  ellipse  about  its 

niajiir  axis.     This  is  the  prolate  spheroid. 

6.  Find  ((juation  of  surface  generated  by  revolution  of  ellipse  about  its 
minor  axis.     This  is  the  oblate  spheroid. 

o 


194 


A  NA  L  YTIC  GEOMETK  Y 


1.    Find  equation  of  surface  generated  by  revolution  of  hyperbola  about 
its  conjugate  axis.     This  is  the  hyperboloid  of  revolution  of  one  sheet. 

8.   Find  equation  of  surface  generated  by  revolution  of  hyperbola  about 
its  transverse  axis.    This  is  the  hyperboloid  of  revolution  of  tv?o  sheets. 

9.  Let  PP'  be  perpendicular 
to  the  JT-axis,  but  not  in  the 
ZX-plane.  Suppose  PP'  to  re- 
volve about  the  Z-axis.  The 
equation  of  the  surface  gener- 
ated is  to  be  found. 

The  equations  of  the  projec- 
tions of  PP'  on  the  planes  ZX 
and  ZY  are  x  =  a.,y  =  bz.  The 
point  P  describes  the  circum- 
ference of  a  circle  whose  equa- 
tion is  x2  +  ^2  _  ,.2.  The  value 
i  of  r  depends  on  z,  and  from  the 

^^«- 1'^"-  figure   r''=  a?+  Vh'K      Hence 

the  equation  of  the  surface  generated  is  x-  +  y'^  =  &%-  -f  a'-.     The  surface 
is  thei'efore  an  hyperboloid  of  revolution  of  one  sheet. 


Akt. 


The  Ellipsoid 


In  the  XF-plane  there  is  the  fixed  ellipse  ^-f--^  =  1,  in  the 
.2         2  «'       ^ 

ZX-plane  the  fixed  ellipse  -  +  -"  =  1.  The  figure  generated 
a-  c- 
by  the  ellipse  which  moves  with 
its  center  on  the  X-axis,  the  plane 
of  the  ellipse  perpendicular  to  the 
X-axis,  the  axes  of  the  ellipse  in 
any  position  the  intersections  of 
the  plane  of  the  ellipse  Avith  the 
fixed  ellipses,  is  called  the  ellip- 
soid.   The  equation  of  the  ellipse 

From  the  equations  of   the  fixed   ellipses 


C Uli  \ 'ED   S  URFA  CES 


195 


— +--  =  1,  —  +  --=1,  whence  rs  =lrll-      \,rt—c-(l ]• 

a-     b-  a-     c-  \       a-j  \       a-j 

Hence  tlie  equation  of  the  generating  ellipse  in  any  position, 

that  is,  the  e(uiation  of  the  ellipsoid,  is  —  +  4,  +  — ,  =  1-     When 

(r      b-     & 
a,  b,  c  arc  unequal,  the  figure  is  an  ellipsoid  with  unequal 
axes;  when  two  of  the  axes  are  equal,  the  figure  is  an  ellipsoid 
of  revolution,  or  spheroid;  Avhen  the  three  axes  are  equal,  the 
elli[)soid  becomes  the  sphere. 


Art. 


The  Hyperboloids 


In  the  ZX-plane  there  is  the  fixed  hyperbola  —  — r,  =  1,  in 
the  Zl'plane  the  fixed  hyperbola  ^  — ^, 


cr 
1.     The  figure  gen- 

Z 


crated  by  the  ellipse  which 
moves  with  its  center  on 
the  Z-axis,  the  plane  of  the 
ellipse  perpendicular  to  the 
Z-axis,  the  axes  of  the  el- 
lipse in  any  position  the 
intersections  of  the  plane 
of  the  ellipse  with  the  fixed 
hyperbolas,  is  called  the 
hyperboloid  of  one  sheet. 
The  equation  of  the  ellipse  Fig.  i7t. 

1.     From  the  equations  of  the  fixed  hyperbolas 


rs'      rt 


ri _?'=:!,    ^-^=1,    whence    ^s'=a-(l  +  '^,  7r=b'(l+''\ 
rr     c-  b'      c-  '  \       c-J  \       c-J 

Hence  the  equation  of    the   generating   ellipse   in    any  posi- 
tion, that  is,  the  equation  of  the  hyperboloid  of  one  sheet,  is 

a-      b-      c-  yS     ^2 

In  the  ZX-plane  there  is  the  fixed  hyperbola  — — 0  =  ^^ 


196 


ANALYTIC  GEOMETRY 


in  the  XT-plane  the  fixed  hyperbola  —  —  ^-=^1.      The  figure 

a^     h- 

generated  by  the  ellipse  which  moves  with  its  center  on  the 

X-axis,  the  plane  of  the  el- 
^  ''  lipse    perpendicular    to    the 

X-axis,  the  axes  of  the  el- 
lipse in  any  position  the  in- 
tersections of  the  plane  of 
the  ellipse  with  the  fixed 
hyperbolas,  is  called  the 
hyperboloid  of  two  sheets. 
The  equation  of  the  ellipse 

is   ^  +  |r,=  l-      Fi'oi^   tlie 

Fig.  172.  '>'^         ^'^ 

9        — 2  o         -p 

equations   of   the   fixed   hyperbolas    —  —  -—-=1,    — -^—^1 

whence  ri^h'{-^-\\,  '^V^  (?(^^-\\.     Hence  the  equation 


of  the  generating  ellipse  in  any  position,  that  is,  the  equation 


of  the  hyperboloid  of  two  sheets,  is  —  ■ 


Art.  100.  —  The  Paraboloids 

In  the  XF-plane  there  is  the  fixed  parabola  -if'  —  2  hx,  in  the 
ZX-plane  the  fixed  parabola  z^=2  ex.  The  figure  generated 
by  an  ellipse  which  moves  with  its 
center  on  the  X-axis,  its  plane  per- 
pendicular to  the  X-axis,  the  axes 
of  the  ellipse  in  any  position  the 
intersections  of  the  plane  of  the 

X     ellipse  with  the  fixed  parabola,  is 

called  the  elliptical  paraboloid.  The 
equation  of  the  ellipse  is 

rs      rt 


CUR  VED  S  UIIFA  CES 


11)7 


From  the  equations  of  the  fixed  paraboUis  rs'  =  2  bx,  rt  =  2  ex. 
Hence  the  equation  of  the  generating  ellipse  in  any  position, 

that  is,  the  e(iuation  of  the  elliptical  paraboloid,  is  •—-{ —  =  2x. 

b       c 

In  the  ZX-plane  there  is  the  fixed  parabola  2-  =  2  ex,  in 
the  Xl'-plane  the  fixed  parabola  y^  —  —  2bx.  The  figure  gener- 
ated by  an  hyperbola  which 
moves  with  its  center  on  the 
X-axis,  the  plane  of  the  hy- 
])erbola  perpendicular  to  the 
X-axis,  the  axes  of  the  hy- 
l)erbola  the  intersections  of 
the  plane  of  the  hyperbola 
with  the  fixed  parabolas,  is 
called  the  hyperbolic  para- 
boloid. The  equation  of  the 
hyperbola  is 


=  1. 


z'  _  y 

rs       rt' 
From  the  equations  of   the 
fixed    parabolas    rs   =  2  ex, 
ri'=—2bx.    Hence  the  equa- 


FiG.  174. 


tion  of  the  generating  hyper 

bola  in  any  position,  that  is,  the  equation  of  the  hyperbolic 

paraboloid,  is —=2x. 

c      b 


Art.  101, —The  Conoid 


The  center  of  an  ellipse  moves  in  a  straight  line  perpendicu- 
lar to  the  plane  of  the  ellipse.  The  major  axis  is  constant  for 
all  positions  of  the  ellipse,  the  minor  axis  diminishes  directly 
as  the  distance  the  ellipse  has  moved,  becoming  zero  when  the 


108 


ANALYTIC  GEOMETRY 


ellipse  has  moved  the  distance  c.  The  figure  generated  is  called 
the  conoid  with  elliptical  base. 
The  equation  of  the  ellipse  is 


y- 


where 


4-^^  —  1 
-::  ^  — 2  —  -^' 
s       rt 

a,  and,  from  similar 

z 


triangles,  —  = -,  whence  rt 

b         c 

-(c  —  z).     The   efjuation   of   the 
c 

generating  ellipse   in   any   posi- 
tion, that  is,  the  equation  of  the 


conoid,  is 


lP{c 


Art.  102. 


Surfaces  represented  by  Equations  in 
Three  Variables 


An  equation  ffi{x,  ?/,  z)=0,  when  intei-preted  in  rectangular 
space  coordinates,  represents  some  surface.  For  when  z  is  a 
continuous  function  of  x  and  y,  if  (x,  y)  takes  consecutive  posi- 
tions in  the  XF-plane,  the  point  {x,  y,  z)  takes  consecutive 
positions  in  space.  Hence  the  geometric  representation  of  the 
function  </>  (x,  y,  «)  =  0  is  the  surface  into  which  this  function 
transforms  the  XF-plane.  To  determine  the  form  and  dimen- 
sions of  the  surface  represented  by  a  given  equation,  the  inter- 
sections of  this  surface  by  planes  parallel  to  the  coordinate 
planes  are  studied. 

Problems.  —  Determine  the  form  and  dimensions  of  the  surfaces  rep- 
resented by  the  following  equations. 

1.   E--|-^-)-  ^2—  \     Tiie  equation  of  the  projection  on  the  Xr-plane 
9       4 
of  the  intersection  of  the  surface  represented  by  this  equation  and  a  plane 

g  =  c  parallel  to  the  AT-plane  is  ^  +  ^  =  1  -  c-.    This  equation  repre- 


CURVED   SURFACES 


199 


sents  an  ellipse  whose  dimensions  are  greatest  when  c  =  0,  diminish  as  the" 
numerical  value  of  c  increases  to  1,  and  are  zero  when  c  =  ±  1.  The 
ellipse  is  imaginary  when  c  is  numerically  greater  than  1. 

The  equation  of  the  projection  on  the  ZA'-plane  of  the  intersection  of 

the  surface  by  a  plane  tj  =  b  parallel  to  the  ZX-plane  is  — \-  z'^=  I  —  — 

9  4 

which  represents  an  ellipse  whose  dimensions  are  greatest  when  ft  =  0, 
diminish  as  b  increases  numerically  to  2,  are  zero  when  ft  =  ±  2,  and 
become  imaginary  when  ft  is  numerically  greater  than  2. 

The  equation  of  the  projection  on  the  TZ-plane  of  the  intersection  of 


the  surface  by  a  plane  x 


a  parallel  to  the  FZ-plane  is  ^  +  2^ 
4 


a2 


which  represents  an  ellipse  whose 
dimensions  are  greatest  when  a  =  0, 
diminish  as  a  increases  numerically 
to  3,  are  zero  when  a  =  ±  3,  and  be- 
come imaginary  when  a  is  numeri- 
cally greater  than  3. 

The  sections  of  the  surface  made 
by  planes  parallel  to  the  coordinate 
planes  are  all  ellipses,  the  surface  is 
closed  and  limited  by  the  faces  of 
the  rectangular  parallelopiped  whose 
faces  are  x  —  ±S,  y  =  ±2,  z  =  ±1. 
From  the  equation  it  is  seen  that  the  origin  is  a  center  of  symmetry,  the 
coordinate  axes  are  axes  of  symmetry,  the  coordinate  planes  are  planes  of 
symmetry  of  the  surface.     The  figure  is  the  ellipsoid  with  axes  3,  2,  1. 

0. 

10x  =  0. 


Fig.  176. 


X-  +  2/-^  -  z^ 
x"^  +  >j-  +  z"^ 
r/  +  z^  ■ 

X2  +  J/2  . 


10x  =  0. 

^2:^1. 


z'  -  2  X  +  o  !i 

j2  +  2  X  +  4  i'  : 


12.    Show  that  the  conoid  is  a 
ruled  surface. 


CHAPTER   XV 

SECOND  DEGEEE  EQUATION  IN  THKEE  VAEIABLES 

Art.  103.  —  Transformation  of  Coordinates 

Take  the  point  (a,  b,  c)  referred  to  the  axes  X,  Y,  Z  as  the 
origin  of  a  set  of  axes  X',  Y',  Z'  parallel  to  X,  Y,  Z  respec- 
tively. Let  (.r,  y,  z),  {x',  y',  z')  represent  the  same  point  referred 
to  the  two  sets  of  axes.    From  the  figure  x  =  x'  +  a,  y  =  y'-\-b, 

z  =  z'  -\-  c. 

z' 


Let  X,  F,  Z  be  a  set  of  rectangular  axes,  X\  Y\  Z  any  set 
of  rectilinear  axes  with  the  same  origin.  Denote  the  angles 
made  by  A''  with  A,  F,  Z  by  a,  (3,  y  respectively,  the  angles 
made  by  Y'  with  A,  Y,  Zhj  a',  y8',  y',  the  angles  made  by  Z' 
with  A,  Y,  Z  by  a",  j8",  y".  If  (a-,  y,  z)  and  (x',  y',  z')  represent 
the  same  point  F,  x  is  the  projection  of  the  broken  line 
200 


SECOND  DEGREE  EQUATION  201 

(x'  +  If'  +  z')  on  tlio  X-axis,  y  the  projection  of  this  broken  line 
on  the  i'-axis,  z  the  projection  of  this  broken  line  on  the 
Z-axis.     Hence 

X  =  x'  cos  a  -\-  ?/'  cos  a'  +  z'  cos  a", 

y  =  x'  cos  f^  +  ?/'  cos  /3'  +  z'  cos  /3", 
Z  =  x'  COS  y  -f-  )/'  COS  y'  +  ;<;'  cos  y". 

Since  X,  F,  Z  are  rectangular  axes, 

cos^  a  +  cos^  /?  +  cos^  y  =  1 , 

COS-«'+  C0S-/5'+  COS-y'  =  1, 

cos^«"  +  cos-;8"-f  cos^y"  =  1. 
If  X',  I"',  Z'  are  also  rectangular, 

cos  a  cos  a'  -f  cos  ft  cos  /?'  +  cos  y  cos  y'  —  0, 
cos  «  cos  a"  +  cos  /3  cos  (3"  +  cos  y  cos  y"  =  0, 
cos  a'  cos  «"  -f  cos  ft'  cos  /3"  +  cos  y'  cos  y"  =  0. 

Problems.  —  1.  Transform  x-  +  y^  +  z-  =  2G  to  parallel  axes,  origin 
(-5,0,0). 

2.  Transform  x-  +  7j-  +  z-  =  25  to  parallel  axes,  origin  (  —  5,  -  5,  —  5). 

3.  Transform  ^  +  ^  -f  IT  =  1  to  parallel  axes,  origin  (  -  a,  0,  0). 

a'^      b-     c^ 

4.  Show  that  the  first  degi-ee  equation  in  three  variables  interpreted 
in  oblique  coordinates  represents  a  plane. 

5.  Show  that  the  equation  of  an  elliptic  cone,  vertex  at  origin,  and 

3.2  y2  5-2 

axis  the  Z-axis,  is  ~  +  ^ —  —  =  0. 
'      a^      62     c-i 

6.  Derive  the  formulas  for  transformation  from  one  rectangular  sys- 
tem to  another  rectangular  system,  the  Z'-axis  coinciding  with  the  Z-axis, 
the  X'-axis  making  an  angle  d  with  the  A'-axis. 

Art.  104. —  Plane  Section  of  Quadric 

Surfaces  represented  by  the  second  degree  equation  in  throe 
variables 
Ax"^  +  By-  +  Cz-  +  2  Dxy  +  2Exz  +  2  Fyz 

*  ■^2Gx  +  2Hy  +  2Kz  +  L  =  0        (1) 
are  known  by  the  general  name  of  quadrics. 


202  ANALYTIC  GEOMETRY 

To  find  the  intersection  of  the  surface  represented  by  this 
equation  by  any  plane  transform  to  a  set  of  axes  parallel  to 
the  original  set,  having  some  point  (a,  b,  c)  in  the  cutting  plane 
as  origin.     The  transformation  formulas  are 

x  =  x'  -\-a,         y  =  y'  -\-  b,         z  =  z'  -\-c, 

and  the  transformed  equation  is 

Ax''  +  By''  +  Cz"  +  2  D'x'tj'  +  2  E'x'z' 

+  2  F'y'z'  +  2  G'x'  +  2  H'y'  +  2  K'z'  +  L'  =  0,      (2) 

where  G'  =  .la  +  Db  +  Ec  +  G, 

H'  =  Bb  +  Da  +  Fc  +  H,  K'  =  Cc  +  Ea  +  Fb  +  K, 

L'  =  Aa'  +  Bb-  +  Cc-  +  2  Dab  +  2  Eac 

+  2  F6c  +  2  (^a  +  2  /f  &  +  2  /ic  +  L. 

Now  turn  the  axes  X',  Y',  Z'  about  the  origin  until  the 
X'  F'-plane  coincides  with  the  cutting  plane.  This  is  accom- 
plished by  the  transformation  formulas 

x'  =  Xi  cos  a  +  ?/i  cos  «'  +  Zi  cos  a", 
y'  =  Xi  cos  y8  +  v/i  cos  /5'  +  ^1  cos  |8", 
2'  =  .Tj  cos  y  +  ?/i  cos  y'  +  2:1  cos  y". 

These  formulas  are  linear,  hence  the  equation  of  the  quadric 
in  terms  of  {x^,  y^,  Zj)  is  of  the  form 

A,x,'  +  B,y,'  +  C,z,'  +  2  D,x,y,  +  2  E,x,z, 

+  2  jPj^^i^i  +  2  (^^.Ti  +  2  if,y,  +  2  /r.^i  +  Xi  =  0.     (3) 

Since  the  plane  of  the  section  is  the  Xj^Vplane,  the  equa- 
tion of  the  intersection  referred  to  rectangular  axes  in  its  own 
plane  is  A.x^'  -f  JB,2/i'  +  2  D.x.y^  +  2  G,x,  +  2  H,y,  +  L,=  0, 
which  represents  a  conic  section.  Hence  every  plane  section 
of  a  quadric  is  a  conic  section.  For  this  reason  quadrics  are 
also  called  conicoids. 


SECOND   DEGIIEE  EQUATION 


20'.] 


Art.  105.  —  Ckxtku  of  Quadric 

The  surface  represented  by  eq\iation  (2)  is  symmetrical  with 
respect  to  the  origin  (a,  b,  c)  wlien  the  coefficients  of  x',  y',  z' 
are  zero,  for  then  if  {x\  y',  z')  is  a  point  of  the  surface, 

(-•<  -y',  -2') 

is  also  a  point  of  the  surface.    Hence  the  center  of  the  quadrie 
(1)  is  found  by  solving  the  equations 

Aa+  Db  +  Ec  +  G  =  0,  Bb  +  Da  +  Fc  +  11  =  0, 

and  Cc  +  Ea  +  Fb  +  K=  0. 

Problems.  —  1.    Find  the  center  of  the  quadrie  represented  by 
a:2  +  ?/2  ^  4  ^2  _  8  a;5:  +  0  2/  =  0. 

2.    Find  the  center  of  the  quadrie  represented  by 

.r2  -  2/2  4-  2;2  -  10  X  +  8  2  +  15  =  0. 


Art,   106.  —  T.vxgent  Plane  to  Quadric 

Let  (.To,  ?yo>  2:0)  be  any  point  of  the  quadric  (1).  The  equa- 
tions cc  =  .To  +  d  cos  (it.,y  =  ?/o  +  d  cos  p,z  =  Z(i-\-  d  cos  y  represent 
all  straight  lines  through  (.t,„  ?yo>  ^^-     By  substituting  in  (1) 

=  0 


+  ^aV 

+  2  oleosa 

a'o 

d+^lcos-« 

+  By,' 

+  2Bcos^ 

2/0 

+  5cos^/8 

+  Cz,? 

+  2  Coos  y 

2^0 

+  C'cos-y 

+  2Z>.r,,Vo 

+  2I>cos« 

•Vo 

+  2  Z>  cos  a  cos  /? 

+  2Ex^, 

+  2  Z)  cos  ^ 

•Ty 

+  2  £"  cos  a  cos  y 

+  2FyoZo 

+  2£oosy 

.To 

+  2i^C0S)8C0Sy 

+  2Gx, 

+  2£cosa 

2^0 

+  2  By, 

+  2Fcosy 

2/0 

+  2Kz, 

+  2FcoS|8 

2^0 

+  L 

+  2  (7  cos  a 
+  2//cos^ 
+  2  A"  cosy 

204  ANALYTIC  GEOMETRY 

an  equation  is  found  which  determines  the  two  values  of  d 
corresponding  to  the  points  of  intersection  of  straight  line  and 
quadric.  Since  the  point  (a^o,  yo,  z^)  lies  in  the  quadric,  the 
term  of  this  equation  independent  of  d  vanishes.  If  the  co- 
efficient of  the  first  power  of  d  also  vanishes,  the  equation  has 
two  roots  equal  to  zero ;  that  is,  every  straight  line  through 
the  point  (;Xq,  ?/„,  ^o),  and  whose  direction  cosines  satisfy  the 
equation 

A  cos  a  •  x^+B  cos  (3  •  ?/o+  C'cos  y  •  z^,-\-D  cos  a  •  ?/(,  +  Z)  cos  /5  •  .Tq 
-f^cosy  •  Xo+Ecosa  ■  Z(t-\-Fcosy  ■  y^+Fcos/B  •  Zq 
-f  GrCos«  +  HcosfS  -f  A'cos  y  =  0 

is  tangent  to  the  quadric.  To  determine  the  surface  repre- 
sented by  this  equation  multiply  by  d  and  substitute  x  —  Xq  for 
d  cos  a,  y  —  ?/o  for  dcos  (3,  z  —  z^  for  d  cos  y.  There  results  the 
equation 

AxX(,  +  Byyo  -f  Czz^  +  D  (;xy^,  +  x^y)  +  E  (xz^  +  x^) 

+  F(yZo  +  y,z)  +  G(x  +  x^)  +  H{y  +  y,)  +  K{z  -f  ^o)  +  ^  =  0, 

which,  since  it  is  of  the  first  degree  in  {x,  y,  z)  represents  a 
plane.  This  plane,  containing  all  the  straight  lines  tangent  to 
the  quadric  at  {xq,  y^,  Zq)  is  tangent  to  the  quadric  at  (.Tu,  ?/„,  z^^. 
Notice  that  the  equation  of  the  plane  tangent  to  the  quadric  at 
(xq,  yo,  Zq)  is  obtained  by  substituting  in  the  equation  of  the 
quadric  xxq  for  x^,  yy^  for  y^,  zz,,  for  z^,  xy^  +  x^y  for  2  xy, 
xZq  +  X(^  for  2  xz,  yzo  -f-  y^z  for  2yz,  x  -\-  x^  for  2x,  y  -\-  y^  for  2  y, 
z  +  Zq  for  2  z. 

Let  (x',  y',  z')  be  any  point  in  space,  (x^,  yo,  Zq)  the  point  of 
contact  with  the  quadric  (1)  of  any  plane  through  (x',  y',  z') 
tangent  to  the  quadric.  Then  (Xq,  y^,  z^,  (x',  y',  z')  must  satisfy 
the  equation 

.4.r'.T„  +  By'yo  +  Cz'z^  -f  D  (x'y,  +  ?/'a'„)  -f  ^(^'.^o  -f  .x-'^o) 

-f  F(z'y,  +  y'z,)  +  G  (;«'  +  x^)  +  U{y'  -{-  y^)  +K{z'  +  z,)  +L  =  0. 


SECOND   DKCREK   EQUATION  205 

Hence  the  points  of  contact  (.t,„  ?/„,  z^■)  must  lie  in  a  ]>lane, 
and  the  locus  of  the  points  of  contact  is  a  conic  section. 

Problems.  —  1.    Write  the  equation  of  the  phiiie  tangent  to 
.T-  +  y-  +  z^  =  B^  at  (xo,  2/0,  Zu). 

2.  Write  the  ciiuation  of  the  plane  tangent  to 

a:2  +  y2  4.  ^2  _  10  a;  +  25  =  0  at  (5,  0,  0). 

3.  Write  the  equation  of  the  plane  tangent  to 

^  +  ?^'  +  -'=lat  (Xo,  2/0,  20). 
a^     62     c- 

4.  Write  the  equation  of  the  plane  tangent  to 

t^?l  =  2xat  (Xo,  2/0,  2o). 
b       c 

5.  Find  equations  of  projections  on  planes  ZX  and  ZY  of  locus 
of  points  of  contact  of  planes  tangent  to  x-  +  y"^  +  z"^  —  25  through 
(7,  -  10,  6). 

6.  Find  equation  of   normal  to— +  ^  +  — =1    at   (x',ij',z').     The 

a^     b'^     c^ 
normal  to  a  surface  at  any  point  is  the  line  through  that  point  perpen- 
dicular to  the  tangent  plane  at  that  point. 

7.  Find  the  angle  between  the  normal  to  —  +  ^^-{-^=1  at  (x',  y',  z') 

cfi     h^     c'^ 
and  the  line  joining  (x',  y',  z')  and  the  center  of  the  ellipsoid. 


Art.  107.  —  Reduction  of  General  Equation  of  Quadric 

To  determine  the  form  and  dimensions  of  the  surfaces  repre- 
sented by  the  general  second  degree  equation  in  three  variables 
when  interpreted  in  rectangular  space  coordinates  it  is  desirable 
first  to  simplify  the  equation.  This  simplification  is  effected 
by  changing  the  position  of  the  origin  and  the  direction  of  the 
axes. 

The  change  of  direction  of  rectangular  axes  is  effected  by 
the  formulas 

X  =  x'  cos  a  +  y'  cos  «'  +  ^'  cos  a", 
y  =  x'  cos  (3  ■{-  y'  cos  /3'  +  2'  cos  /3", 
z  =  a;'  cos  y  4-  y'  cos  y'  -f-  z'  cos  y", 


206  ANALYTIC  GEOMETRY 

where  the  nine  cosines  are  subject  to  the  six  conditions 
cos^  a  +  cos^  /3  +  cos-  y  =  1 , 
cos'^w'  +  cos^)8'  +  cos-y'  =  1, 

COS^«"  +  C0S-/3"  +  COS-y"  =  1, 

cos  a  cos  «'  +  COS  (3  cos  (3'  +  cos  y  cos  y'  =  0, 
cos  a  cos  «"  +  cos  (3  cos  (3"  +  cos  y  cos  y"  =  0, 
cos  a'  COS  a"  +  cos  yS'  cos  /3"  +  cos  y'  cos  y"  =  0. 
Three  arbitrary  conditions  may  therefore  be  imposed  on  the 
nine  cosines. 

Substituting  for  x,  y,  z  in 

Ax"  +  By-  -^Cz--^2  Dxy  +  2  Exz  +  2Fyz  +  2Gx 
+  2Hy-^2  Kz  +  L  =  0, 
there  results 

Ax"  +  By"  +  Cz'-  +  2  D'x'y'  +  2  ^'a-'^'  +  2  F'?/'^'  +  2  G'x' 

+  2H'y'  +  2K'z'  +  L'  =  0, 

when  D',  E',  F'  are  functions  of  the  nine  cosines.  Equate 
B',  E',  F'  to  zero  and  determine  the  directions  of  the  rectangu- 
har  coordinates  in  space  in  accordance  with  these  equations. 
This  transformation  is  always  possible,  hence 

Ax'  +  Bf-  +  Cz'  +  2  G'x  +  2  IFy  +  2  K'z  +  L'  =  0 

interpreted  in  rectangular  coordinates  represents  all  quadric 
surfaces. 

Now  transform  to  parallel  axes  with  the  origin  at  (a,  b,  c). 
The  transformation  formulas  are 

X  =  a  -It  x',  y  =  h  +  y',  z  =  c  +  z' 

and  the  transformed  equation 

Ax"+By"  +  Cz"  +  2(Aa  +  G')x'  +  2(Bb  +H')y'+2(Cc  +  K')z' 

+  (Aa'  +  BW  +  Cc-  +  2  G\i  +  2  //7>  +  2  K'c  +  L')  =  0. 


SECOND  DEGREE  EQUATION  207 

Take  advantage  of  the  three  arbitrary  constants  a,  b,  c  to  cause 
the  vanishing  of  the  coefficients  of  x',  y',  z'.     This  gives 

=  -^    b  =  -—     '  =  -=^ 
"  A'  B'   ''  C' 

values  whicli  are  admissible  when  .1  ^^  0,  B  :^  0,  C  ^  0.     The 

resulting  equation  is  of  the  form  Lx'  +  3Ii/  +  iVV  =  P. 

When  A  ^  0,  B  ^  0,  C  =  0,  the  transformation 

x  =  a  +  x',y  =  h-\-y',z  =  c  +  z' 
gives 

Ax"  +  By"  +  2(Aa+  G)x>  +  2(Bb  +  H')y'  +  K'z' 

+  (Aa'  +  Bb'  +  2  G'a  +  2H'b  +  2  K'c  +  L')  =  0. 

Equating  to  zero  the  coefficients  of  x',  y'  and  the  absolute  term, 

the  values  found  for  a,  b,  c  are  finite  when  A^O,  -B  =5^  0,  K'  =^  0. 

The  resulting  equation  is  of  the  form  Lx?  +  My-  +  N'z  =  0. 

When  A^O,  B^O,  C  =  0,  K'  =  0,  the  equation  takes  the 
form  LiT  +  My-  +  X'x  +  M'y  +  P  =  0. 

When  url  ^  0,  iJ  =  0,  C  =  0,  the  equation  takes  the  form 
Mx""  +  M'x  +  N'y  +  P  =  0. 

When  A  =  0,  B  —  0,  C  —  0,  the  equation  is  no  longer  of  the 
second  degree. 

Since  x,  y,  z  are  similarly  involved  in 

Ax"  +  By-  +  0x^  +  2  G'x  +  2  IVy  +  2  K'z  +  L'  =  0, 
the  vanishing  of  A  and  G'  or  of  B  and  H'  would  lead  to  equa- 
tions of  the  same  form  as  the  vanishing  of  C  and  7i '. 

Collecting  results  it  is  seen  that  the  following  equations 
interpreted  in  rectangular  coordinates  represent  all  quadric 
surfaces  — 

^1:^0,  B^Q,  C4-(),  Lx'+My-  +  Nz-=P  I 

A^^O,  B^O,  C=0,  K'^0     Lx-+My''+N'z=0  II 

A^O,  B^O,  C=0,  K'  =  0     Lx^-^My^+M'y-{-L'x+P=0\ 
A^O,  B=0,  C=0  Lx^+L'x-{-M'y+N'z  +  P=0\ 

These  equations  are  known  as  equations  of  the  first,  second, 
and  third  class. 


208  ANALYTIC  GEOMETRY 

Art.  108.  —  Sukfaces  of  the  First  Class 

The  equation  of  the  first  class  may  take  the  forms 
(a)  ix-  +  31  f  +  Nz^  =  P,     (b)  Lx"  +  3Iy-  -  Nz-  =  P, 
(c)  Lx^  +  3bf-Nz'^  =  -P, 

or  similar  forms  with  the  coefficients  of  ar  and  z-  or  of  'if  and  z' 
positive. 

(a)  The  intersections  of   planes  parallel  to  the  coordinate 
planes  with  Lx^  +  3Iy/  +  Nz'  =  P  are  for 

X  =  x',  3Iif  +  Nz~  =  P-  Lx'\ 
an  ellipse  whose  dimensions  are  greatest  when  x'  =  0,  diminish 
as  x'  increases  numerically,  are  zero  for  x'  =  ±  \-jr,  imaginary 

—  '  X/ 

— ; 

for  y  =  ?/',  Lx-  +  Nz-  =  P  —  My'-, 

an  ellipse  whose  dimensions  are  greatest  when  _?/'  =  0,  diminish 
as  y'  increases  numerically,  are  zero  for  ?/'  =  ±a/— ,  imaginary 
when  y'  is  numerically  greater  than  -v/ —  ; 

for  z  =  z',  Lx-  +  3ry-  =  P  —  Nz'-, 

an  ellipse  whose  dimensions  are  greatest  for  z'  =  0,  diminish 

as  z'  increases  numerically,  are  zero  when  z'  =  ±\^,  imaginary 
when  z'  is  numerically  greater  than  a  /  -  • 

Calling  the  semi-diameter  on  the  X-axis  a,  on  the  I''-axis  b, 


on  the  .^-axis  c,  the  equation  becomes  — -f^  +  — =  1,  the 
ellipsoid.  "       ^       ^ 

The  figure  represented  by  Lx^ -\- 3Df  +  Nz^  =  —  P  is  imagi- 
nary.    The  equation  Lx-  +  3[y-  +  Nz-  =  0  represents  the  origin. 

(b)  Lx^  -\-  3fy^  —  Nz^  =  P.     The  intersections  are  for  x  =  x', 


SECOND   DKGRKE  EQUATION  201) 

3/)/-  —  Xz-  =  r  —  Lx'-,  ;iu  hyperbola  whose  real  axis  is  parallel 
to  the  I'-axis  when  — \/.  <-<^'<  +\-,>  parallel  to  the  Z-axis 

when  x'  is  numerically  greater  than  -il  -,  and  which  becomes 

two  straight  lines  Avhen  x'  =  ±-1/—; 
'  L 

for  !/  =  y',  Lx""  -  .V^-  =  P-  My", 

an  liy}ierbola  whose  real  axis  is  parallel  to  the  X-axis  when 

parallel   to   the   Z-axis   when   y'  is  numerically  greater  than 

(/>  jp 

\/— ,  and  which  becomes  two  straight  lines  when  ^' =  ±\/— : 
V  M  ^  M 

for  z-z',  Lx' +  My- =  P  +  Nz'-, 

an  ellipse,  always  real,  whose  dimensions  are  least  when  z'  =  0, 
and  increase  indefinitely  when  z'  increases  indefinitely  in  nu- 
merical value.  Calling  the  intercepts  of  this  surface  on  the 
X-axis  a,  on  the  F-axis  b,  on  the  Z-axis  cV— 1,  the  equation 

becomes  —  -f  -'^  —  -  =  1,  the  hyperboloid  of  one  sheet. 
a^     b'     & 

(c)  Lx'  -f  My'  -Nz'  =  -  P. 

The  intersections  are 

for  X  =  x',  My'  -Nz'  =  -P-  Lx", 

an  hyperbola  with  its  real  axis  parallel  to  the  Z-axis,  dimen- 
sions least  when  a;'  =  0,  increasing  indefinitely  with  the  numeri- 
cal value  of  x' ; 

for  y  =  y',  Lx"  -Nz'  =  -  P-  My", 

an  hyperbola  with  its  real  axis  parallel  to  the  Z-axis,  dimen- 
sions least  when  y'  —  0,  increasing  indefinitely  with  the  numeri- 
cal value  of  y' ; 
for  z  =  z',  Lx'  +  My-  =  Lz"  -  P, 


210  ANALYTIC  GEOMETRY 


an   ellipse,    imaginary   when 


—a/—  <  2;' <  -\-\y'    dimensions 
zero  for  z'  —  ±\j—,  increasing  indefinitely  with  the  numerical 

value  of  z'. 

Calling  the  intercepts  of  this  surface  on  the  axes  X,  Y,  Z 
respectively,  aV—  1,  6V—  1.  c,  the  equation  becomes 

a?      Ir     c- 
the  hyperboloid  of  two  sheets. 

The  surfaces  of  the  first  class  are  ellipsoids  and  hyperboloids. 


Art.  109.  —  Surfaces  of  the  Second  Class 

The  equation  of  the  second  class  may  take  the  forms 
(a)  Lx'  +  Ml/  ±  N'z  =  0,         (&)  Lx"  -  Mxf  ±  N'z  =  0. 

(ct)  Lx-  +  3fy-  =  N'z.     The  intersections  are 
for  x  =  x',  Mf-  =  N'z  -  Lx'\ 

a  parabola  Avhose  parameter  is  constant,  axis  parallel  to  Z-axis, 
and  whose  vertex  continually  recedes  from  the  origin ; 
for  y  =  y',  Lx^  =  N'z  -  My", 

a  parabola  whose  parameter  is  constant,  axis  parallel  to  Z-axis, 
and  whose  vertex  continually  recedes  from  the  origin ; 
for  z  =  z',  Lx^  +  3fy-  =  N'z', 

an  ellipse  whose  dimensions  are  zero  for  z'  =  0  and  increase 
indefinitely  as  z'  increases  from  0  to  +  co,  but  are  imaginary 
for2;'<0. 

This  surface  is  the  elliptic  paraboloid.  The  equation 
Lx^  +  My^  =  —  N'z  represents  an  elliptic  paraboloid  real  for 
negative  values  of  z. 

(b)  Lx^  —  My-  =  N'z.     The  intersections  are 
for  x  =  x',  My- =  Lx'- -  N'z, 

a  parabola  of  constant  parameter  whose  axis  is  parallel  to 


SECOyi)  DEGREE  EQUATION 


211 


the  Z-axis  and  whose  vertex  recedes  from  the  origin  as  x' 
increases  numerically ; 

for  y  =  ij',  Lx^  =  N'z  +  My'-, 

a  parabola  of  constant  parameter  whose  axis  is  parallel  to  the 
Z-axis  and  whose  vertex  recedes  from  the  origin  as  y'  increases 
numerically  ; 
for  z  =  z',  Lxr  —  My-  =  N'z', 

an  hyperbola  whose  real  axis  is  parallel  to  the  X-axis  when 
z  >  0,  paralh'l  to  the  I'-axis  when  z'  <  0,  and  which  becomes 
two  straight  lines  when  z'  =  0. 

The  surface  is  the  hyperbolic  paraboloid. 

The  surfaces  of  the  second  class  are  paraboloids. 


Art.  110.  —  Surfaces  of  the  Third  Class 

The  equation  Lx^  +  3fy^  +  L'x  +  31 'y  +  P  =  0  does  not  con- 
tain z  and  therefore  represents  a  cylindrical  surface  whose 
elements  are  parallel  to  the 
Z-axis.  The  directrix  in 
the  XF-plane  is  an  ellipse 
Avhen  L  and  3f  have  like 
signs,  an  hyperbola  when 
L  and  31  have  unlike  signs. 

The  surface  represented 
by  the  e(piation 

Lx-  +  L'x  +  31' y 

is  intersected  by  the  A'l"- 
})lane  in  the  parabola 
Lx-  +  L'x  +  3['y  +  P  -  0, 
by    the    ZX-plane    in    the 
parabola 
Lx-  +  L'x  -\-N'z  +  P  =  0, 


212  ANALYTIC   GEOMETRY 

by  planes  x  =  x'  parallel  to  the  I'Z-plane  in  parallel  straight 

lines 

N'y  +  L'z  +  Mx'-  +  3I'x'  +  P  =  0. 

Hence  the  surface  is  a  parabolic  cylinder  with  elements  parallel 
to  the  ZF-plane. 

The  surfaces  of  the  third  class  are  cylindrical  surfaces  with 
elliptic,  hyperbolic,  or  parabolic  bases. 

It  is  now  seen  that  the  second  degree  equation  in  three 
variables  represents  ellipsoids,  hyperboloids,  paraboloids,  and 
cylindrical  surfaces  with  conic  sections  as  bases.  Conical  sur- 
faces are  varieties  of  hyperboloids. 

Art.  111.  —  QuADRics  as  Ruled  Surfaces 

The  equation  of  the  hyperboloid  of  one  sheet  '—^ ^  =  1  —  --^ 

is  satisfied  by  all  values  of  x,  y,  z,  which  satisfy  simultaneously 
the  pair  of  equations 

l-l=t.(l-^,     ^  +  ?=.lfl+f\  (1) 

or  the  pair 


a     c         \        bj      a     c     fx 

-     -       ■^+A    -  +  ^=lfl-?A  (3) 


a      c         \        oj     a     c     fjL 


c      a'V        b 


when  fx  and  fx'  are  parameters.  For  all  values  of  /x  equations 
(1)  represent  two  planes  whose  intersection  must  lie  on  the 
hyperboloid.  Likewise  equations  (2)  for  all  values  of  fx'  repre- 
sent two  planes  whose  intersection  must  lie  on  the  hyperboloid. 
There  are  therefore  two  systems  of  straight  lines  generating 
the  hyperboloid  of  one  sheet. 

Each  straight  line  of  one  system  is  cut  by  every  straight  line 
of  the  other  system.  For  the  four  equations  (1)  and  (2)  made 
simultaneous  are  equivalent  to  the  three  equations 


SECOND  DEGREE  EQUATION  21:J 

from  ^vlli{•]l 

//        M  —  /a'       X        1  +  llfx'       Z  _\    —  fjifx.' 


b      /u.  +  /a'     «       /A  +  /a'      c      /a  +  /a' 
Ko  two  straight  lines  of  the  same  system  intersect.     "Write 
the  equations  of  lines  of  the  first  system  corresponding  to  //j 

and  fx.0.     INfaking  the  equations  simultaneous  (/jLi—fi^)!  1  — -- Wo, 

/"l        1  \  /  \  \       ^/ 

and  (- .Yl+?^]=0.      Hence  either  n-i—fx-.,  or  y  =  h  and 


II  =  —  h.  Since  _?/  cannot  be  at  once  +  h  and  —  6,  //.i  =  yu,o ;  that 
is.  two  lines  of  the  same  system  can  intersect  only  if  they 
coincide. 

Observing  that  the  equation  of   the  hyperbolic  paraboloid 

'^  =  2  .T  is  satisfied  by  the  values  of  x,  y,  z,  which  satisfy 

either  of  the  pairsj,  of  equations 


z          y   _  I  X 
Vc      Vb        /*  ' 

Vc      Vb 

0) 

z         y       2x 

Vc      Vb       /' 

Vc      V6 

(-0 

it  can  be  shown  that  this  surface  niaj^  be  generated  by  two 
systems  of  straight  lines ;  that  each  line  of  one  system  is  in- 
tersected by  every  line  of  the  other,  and  that  no  two  lines  of 
the  same  system  intersect. 

The  equations  of  ellipsoid,  hy})erl)()l()id  of  two  sheets  and 
of  elliptical  paraboloid  cannot  be  resolved  into  real  factors  of 
the  first  degree,  consequently  these  surfaces  cannot  be  gener- 
ated by  systems  of  real  straiglit  lines. 

Akt.  112.  —  AsvMi'i'oTic  Sri;FA(;?:s 
From  the  equaticm  of  the  hypcrlxiloid  of  one  sheet 

b- 


214  ANALYTIC  GEOMETRY 


it  is  found  that 


^x^y 


the  powers  of   a^'if  +  h-x^  in  the  denominators  increasing  in 
the  expansion  by  the  binomial  formula.     Hence  the  z  of  the 

hyperboloid  -„  +  ^  —  -  =  1,  and  the  z  of  the  cone 
o}     b^     c- 


•ii  +  ^_  _  ^  =  0 

ce      h"-      & 

approach  equality  as  x  and  y  are  indefinitely  increased ;  that 
is,  the  conical  surface  is  tangent  to  the  hyperboloid  at  infinity. 

■^ —  ^  =  0  is  shown  to  be  asymp- 
a-      h^      <T 

totic  to  the  hyperboloid  of  two  sheets  '-,  —  ^  — ;,  =  1- 

a^     IP-     & 


Art.  113. — Orthogonal  Systems  of  Quadrics 

The  equation  (1 )  — ^ h  tt^^  +  -^r^—  =  1'  where  a>h>c 

and  A  is  a  parameter,  represents  an  ellipsoid  when  co  >  A  >  —  c^, 
an  hyperboloid  of  one  sheet  when  —  c^  >  A  >  —  Ir,  an  hyper- 
boloid of  two  sheets  when  —  6^>A>  — a',  an  imaginary  sur- 
face when  A  <  —  al 

Through  every  point  of  space  {x\  y\  z')  there  passes  one 
ellipsoid,  one  hyperboloid  of  one  sheet,  and  one  hyperboloid  of 
two  sheets  of  the  system  of  quadrics  represented  by  equa- 
tion (1).  Por,  if  A  is  supposed  to  vary  continuously  from 
+  oo  to  —  (X)  through  0,  the  function  of  A, 


a^  -I-  A      b-  +  X      c-  +  A. 


1, 


SECOND    DEGREE   EQUATION  215 

is  —  when  A  = -f  vd  and  +  when  A  is  just  greater  than  —  r, 

—  when  A  is  just  less  than  —  c"  and  +  when  A  is  just  greater 
than  —  //-,  —  wlien  A  is  just  less  than  —  b^  and  again  +  when 
A  is  just  greater  than  —  cr.     Hence 

must  determine  three  real  values  for  A;  one  between  +  oo  and 

—  r,  another  between  —  c-  and  —  b^,  a  third  between  —  b'  and 

—  ((-. 

Let  A„  Ao,  A3  be  the  roots  of  equation  (2) ;  that  is,  let 

g-'-'  y'-  z'-     _  .  -, 

^.'2  -,,»2  ^i-' 

-  1,  (4) 


a-  +  A.,      b'^  +  A,     c-  +  A; 


a-  +  A3      6-  + A3      C-  +  A3        ■  ^'^ 

The  equations  of  tangent  planes  to  the  quadrics  of  system 
(1)  corresponding  to  Ai,  As,  A3  at  the  point  of  intersection 
{x\  y\  z')  are 


XX 

+ 

b'  +  A, 

+  - 

22 

r  +  A, 

=  1, 

xx' 
«■  +  A. 

+ 

b'  +  \, 

+ 

zz' 

=  1, 

xx' 

+ 

+ 

zz' 

=  1. 

((•-  +  A3    b'  +  A3    t-  +  A3 

The  condition  of  perpendicularity  of  the  iii'st  two  [tlanes 

^ +__.'/"'  .+  ^  0 

(a'  +  A,)(«-  +  A.)      {b-'  +  X,){b-'  +  Ao)      (c-  +  AO(r  +  A.) 

is  a  conse(pience  of  (.'i)  and  (4).     In  like  manner  it  is  shown 
that   the   three    tangent   planes   are   mutually  perpendicular. 


216  ANALYTIC  GEOMETRY 

Hence  equation  (1)  represents  an  orthogonal  system  of  quad- 
rics. 

Since  through  every  point  of  space  there  passes  one  ellipsoid, 
one  hyperboloid  of  one  sheet,  and  one  hyperboloid  of  two 
sheets  of  the  orthogonal  system  of  quadrics,  the  point  in 
space  is  determined  by  specifying  the  quadrics  of  the  orthogo- 
nal system  on  which  the  point  lies.  This  leads  to  elliptic 
coordinates  in  space,  developed  by  Jacobi  and  Lame  in  1839, 
by  Jacobi  for  use  in  geometry,  by  Lame  for  use  in  the  theory 
of  heat. 

If  a  bar  kept  at  a  constant  temperature  is  placed  in  a  homo- 
geneous medium,  when  the  heat  conditions  of  the  medium 
have  become  permanent  the  isothermal  surfaces  are  the  ellip- 
soids, the  surfaces  along  which  the  heat  flows  the  hyperboloid s, 
of  the  orthogonal  system  of  quadrics. 


NEW   AMERICAN   EDITION   OF 

HALL  AND  KNIGHT'S  ALGEBRA, 

FOR   COLLEGES  AND  SCHOOLS. 

Revised  and  Enlarged  for  the  Use  of  American  Schools 

and  Colleges. 

By  FRANK   L.  SEVENOAK,  A.M., 

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AMERICAN   EDITION   OF 

ALGEBRA  FOR  BEGINNERS. 

By  H.  S.  HALL,  M.A.,  and  S.  R.  KNIGHT. 

NEVISED    BY 

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